k(n)-torsion-free H-spaces and P (n)-cohomology
J. Michael Boardman W. Stephen Wilson
October 2004
Abstract
In [Wi75 ], for each k, the H-space that represents Brown-Peterson coh*
*omol-
ogy BP k(-) was split into indecomposable factors, which have torsion-fre*
*e ho-
motopy and homology. Here, we do the same for the related spectrum P (n),*
* by
constructing idempotent operations in P (n)-cohomology P (n)k(-) in the s*
*tyle
of [BJW95 ]; this relies heavily on the Ravenel-Wilson determination [RW*
*96 ]
of the relevant Hopf ring. The resulting (i-1)-connected H-spaces Yihave *
*free
connective Morava K-homology k(n)*(Yi), and may be built from the spaces *
*in
the -spectrum for k(n) using only vn-torsion invariants.
We also extend Quillen's theorem on complex cobordism to show that for
any space X, the P (n)*-module P (n)*(X) is generated by elements of P (n*
*)i(X)
for i 0. This result is essential for the work of Ravenel-Wilson-Yagita
[RWY98 ], which in many cases allows one to compute BP -cohomology from
Morava K-theory.
AMS subject classification: Primary 55N22; Secondary 55P45.
Introduction
We exploit the close relationship between the connective Morava K-theory sp*
*ec-
trum k(n), whose coefficient ring is k(n)* = F p[vn], and the spectrum P (n) wi*
*th
P (n)* = F p[vn, vn+1, vn+2, . .]., where Fp denotes the field with p elements.*
* These
ring spectra are defined for each prime p (suppressed from almost all the notat*
*ion)
and integer n 0. Most of our work generalizes the case n = 0 (see [Wi75 ]), w*
*here
k(0) = H(Z (p)), the Eilenberg-Mac Lane spectrum for Z(p)(the integers localize*
*d at
p), and P (0) = BP , the Brown-Peterson spectrum, with BP* = Z(p)[v1, v2, v3, .*
* .]..
In x1, we present three groups of results. First, we give a structure theor*
*em for a
class of H-spaces that may be defined entirely in terms of k(n). Second, starti*
*ng from
P (n), we construct examples of such H-spaces which we use to prove our structu*
*re
theorem. Third, there are consequences for the structure of P (n)-(co)homology:*
* we
find (i) a Quillen-type result, that P (n)*(X) is generated as a module by elem*
*ents
of P (n)i(X) for i 0, (ii) a Landweber-type filtration theorem, and (iii) a b*
*ound on
the homological dimension of P (n)-homology.
All these results depend on the Ravenel-Wilson calculation [RW96 ] of the *
*Hopf
ring for P (n), which encodes the unstable operations in P (n)-cohomology. All*
* the
- 1 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
machinery of [Bo95 , BJW95 ] becomes available, making P (n)*(-) our sixth exa*
*mple
of a cohomology theory whose operations we can handle in a uniform manner.
The authors acknowledge the influence of D. C. Johnson and D. C. Ravenel on
this paper through their work with us on [BJW95 , RW96 ].
Notation We fix throughout a prime p and an integer n > 0. Because it occurs so
frequently, we find it convenient to write N = pn - 1.
[For completeness, we include the results for p = 2. Modifications are req*
*uired
because: (i) our ring spectra are no longer commutative, and (ii) one of our t*
*est
spaces, real projective space, has different cohomology. Shorter comments, like*
* this
one, are enclosed within square brackets. Longer comments form a subsection. A *
*few
proofs are substantial enough to be deferred to a separate paper [Bo ].]
All spaces are assumed to be homotopy-equivalent to cw -complexes. Identity
maps and homomorphisms are denoted by id.
We use much notation and terminology from [BJW95 ]. A ring spectrum E defi*
*nes
a homology theory E*(-) and a cohomology theory E*(-), both multiplicative with
coefficient ring E* = ssS*(E). Then Ei(-) is represented (on the homotopy categ*
*ory
Ho of unbased spaces) by the i-th space E_iof the -spectrum for E.
Because we deal mainly with homology and homotopy groups rather than coho-
mology, we use homology degrees throughout (unlike [BJW95 ]), assigning the de*
*gree
i to elements of Ei(X) and ssi(X). This forces elements of Ei(X) to have degree*
* -i.
We thus write E* for the coefficient ring, even when working with cohomology; in
particular, Ei(point) = E-i. So the Hazewinkel generator vi has degree 2(pi-1).
The algebraic suspension M of a graded group M is a copy of M with all deg*
*rees
raised by one: an element x 2 Mi gives rise to x 2 ( M)i+1.
As in [RW96 ], E(x, . .).denotes the exterior algebra on generator(s) x, *
*. .,.
P (x, . .).the polynomial algebra, and T Ph(x) the truncated polynomial algebra
h
P (x)=(xp ).
1 The main results
Splittings of H-spaces We regard the standard generator uk of P (n)*(Sk) as a
map uk: Sk ! P_(n)_k. We consider spaces X that satisfy the axioms:
(i)X is a connected H-space of finite type (meaning that each homo-
topy group ssi(X) is finitely generated);
(ii)k(n)*(X) is a free k(n)*-module (equivalently, has no vn-torsion);(1.1)
(iii)For any k > 0, any map Sk ! X factors through the map uk to
give a map P_(n)_k! X.
Our first theorem classifies these spaces.
Theorem 1.2 Given n > 0, the spaces X that satisfy the axioms (1.1) have the
following properties:
(a) For each k > 0, there is (up to homotopy) a unique (k-1)-connected (but *
*not
k-connected) example Yk that does not decompose as a product of spaces;
- 2 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Q
(b) Every X is homotopy equivalent to some product Y = iYki, where the num-
ber of copies of each Yk is finite and is uniquely determined by X;
(c) Every retract of X is another example;
(d) Every product of examples is an example, provided it has finite type;
(e) The loop space X is another example, provided X is simply connected.
Shortly, in Definition 1.10, we shall reveal the spaces Yk explicitly.
Remark The above decompositions and equivalences are not as H-spaces. Neverth*
*e-
less, no information is lost, because in (b) for example, the given multiplicat*
*ion on X
corresponds to some multiplication on Y ; as we (shall) have complete informati*
*on on
the possible maps Y x Y ! Y , we can in principle detect which of them are H-sp*
*ace
multiplications.
Part (c) is clear. So is (d), with the help of the K"unneth formula for k(n*
*)-homology
(as in [Bo95 , Thm. 4.2]). Part (e) will follow immediately from (b) and Theore*
*m 1.15.
We prove (a) and (b) in x3.
Towers built from k(n) Although axiom (1.1)(iii) is technically convenient, it
lacks intuitive content. Here, we replace it by a more appealing axiom. This ma*
*kes
Theorem 1.2 analogous to the results of [Wi75 ], as we discuss later in this se*
*ction.
We consider spaces that are built from the spaces k(n)_iin a particularly n*
*ice way,
using only vn-torsion invariants. We recall that k(n)* = Fp[vn], where vn has d*
*egree
2N = 2(pn-1).
Definition 1.3 Given a space Y , we call a map z: Y ! k(n)_q+1a vn-torsion map
if, considered as an element of k(n)*(Y ), it satisfies vcnz = 0 for some c. (W*
*e assume
q 0. Indeed, z must be zero unless q 2N + 1 = 2pn - 1.)
We call a space X a k(n)-tower with vn-free homotopy if it is the homotopy *
*limit
of a sequence of spaces and maps
. .-.-!X3 --! X2 --! X1 --! X0 = point (1.4)
in which each map Xi ! Xi-1 (for i > 0) is the homotopy fibre of some vn-torsion
map zi: Xi-1 ! k(n)_q(i)+1. (We allow the possibility of a finite tower, X = Xm*
* for
some m, or even a tower having only one stage, X = X1 = k(n)_q(1), as well as t*
*he
degenerate case where X is contractible.)
A vn-torsion map z: Y ! k(n)_q+1necessarily induces the zero homomorphism on
homotopy. Then for each i > 0 (assuming X is connected, so that q(i) 1), the
homotopy long exact sequence of zi reduces to the short exact sequence of groups
0 --! q(i)Fp[vn] --! ss*(Xi) --! ss*(Xi-1) --! 0. (1.5)
Thus ss*(X) is an iterated extension of suspensions of Fp[vn]. (Our terminolog*
*y is
abusive to the extent that we do not have a natural action of vn on ss*(Xi) for*
* i > 1.)
We study such towers in more detail in x4 and prove the following equivalen*
*ce.
- 3 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Theorem 1.6 If we replace axiom (iii) in (1.1) by the axiom
(iii)0X is a k(n)-tower with vn-free homotopy, (1.7)
we obtain the same class of H-spaces. Thus Theorem 1.2 remains valid.
Examples based on P (n) The prime ideal
In = (p, v1, v2, . .,.vn-1) BP* = Z(p)[v1, v2, v3, . .].
is invariant and therefore of particular interest. (We set v0 = p, and take I1*
* = (p)
and I0 = (0).) The spectrum P (n) is constructed (see x2) to have the quotient *
*ring
P (n)* = BP*=In = Fp[vn, vn+1, vn+2, . .].
as its homotopy. In particular, P (0) = BP and P (1) is just BP mod p.
Further, given m n, we kill off the ideal
Jm = (vm+1 , vm+2 , vm+3 , . .). P (n)* (1.8)
to produce the spectrum we call P (n, m) (but known to Yosimura [Yo7*
*6 ] as
BP [n, m+1) and to Yagita [Ya76 ] as BP (p, v1, . .,.vn-1, vm+1 , . .).), with *
*homotopy
P (n, m)* = P (n)*=Jm = Fp[vn, vn+1, . .,.vm ].
It comes equipped with a canonical map ae(m): P (n) ! P (n, m). These spectra a*
*re
intimately connected with the spectra E(n, m) = v-1mP (n, m), which are essenti*
*al in
Ravenel-Wilson-Yagita [RWY98 ]. We recognize P (n, n) as k(n).
Remark Unlike In, the ideal Jm is not at all canonical, as it depends on the *
*choice
of the generators viof P (n)*. Nevertheless, our results are independent of thi*
*s choice,
as we are concerned only with the additive structure of P (n, m).
The behavior of these spectra depends on the numerical function
g(n, m) = 2(pn + pn+1 + . .+.pm ), (1.9)
where it is reasonable to define g(n, n-1) = 0.
Definition 1.10 Given k > 0, we define the H-space Yk = P_(n,_m)_k, where the
integer m is defined in terms of equation (1.9) by
g(n, m-1) < k g(n, m). (1.11)
For convenience, we also define Y0 = Fp, viewed as a discrete group.
These are the spaces Yk that appear in Theorem 1.2. In particular, Yk = k(n*
*)_k
for 0 < k 2pn. As the spaces P_(n)_ksatisfy the axioms (1.1), they must decom*
*pose
according to Theorem 1.2(b). We establish the following splittings in x3.
Theorem 1.12 Assume k 0. If k > 0, define m by equation (1.11); if k = 0, *
*take
m = n - 1. Then we have homotopy decompositions
Y
P_(n)_k' Yk x Yk+2(pj-1) (1.13)
j>m
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J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
and, for any h > m,
Yh
P_(n,_h)_k' Yk x Yk+2(pj-1). (1.14)
j=m+1
These are equivalences of H-spaces [except in the extreme case when p = 2 and
k = g(n, m)].
We showed in [BW01 , Thm. 1.1] that such splittings exist, though without *
*making
them explicit as we do here in x3. They are patterned after the splittings of t*
*he spaces
BP__kin [Wi75 ], which were recovered explicitly in [BJW95 ] and are reviewed *
*below.
We note that equation (1.14) reduces to Definition 1.10 when h = m.
Remark No such result holds for P_(n,_m)_kwhen k > g(n, m), as axiom (1.1)(ii)
definitely fails (otherwise this space would contradict Theorem 1.2(b)).
We use equation (1.14) to decompose Yk = P_(n,_m)_k-1explicitly.
Theorem 1.15 The loop space Yk is given for all k > 0 as follows:
(a) If k does not have the form g(n, q) + 1 for any q, then Yk ' Yk-1;
*
* _
(b) If k = g(n, q) + 1, where q n - 1, then Yk ' Yk-1 x Yk-1+2(pq+1-1). *
*|_|
Since is a right adjoint functor and so preserves products, this gives pa*
*rt (e) of
Theorem 1.2. We leave it as an exercise to decompose the negative spaces P_(n)*
*_-k
for k > 0, by writing them as k+1P_(n)_1, and similarly P_(n,_m)_-k.
Some history For n = 0, the results differ slightly. Recall that k(0) = H(Z (*
*p)),
P (0) = BP , and (see [Wi75 ]) P (0, m) = BP has BP * = Z(p)[v1, v2, . .*
*,.vm ].
Axioms (1.1) (with (iii) replaced by (1.7)) then yield connected H-spaces X who*
*se
homotopy groups ssk(X) and homology groups Hk(X) are all free Z (p)-modules of
finite rank. The Postnikov k-invariants of such spaces are necessarily torsion *
*elements.
Theorem 1.2 remains valid exactly as stated, with m still defined by equation (*
*1.11).
However, Theorem 1.12 gives H-space equivalences only for g(0, m-1) < k < g(0, *
*m);
for k = g(0, m), we have merely a homotopy equivalence. (Of course, Y0 = Z(p)ra*
*ther
than Fp.) These are the main results of [Wi75 ] or Theorem 1.16 of [BJW95 ], a*
*nd form
the motivation for this work.
The structure of P (n)-cohomology We extend Quillen's theorem on complex
cobordism to P (n).
Theorem 1.16 For any space X, the cohomology P (n)*(X) is generated, as a
P (n)*-module (topologically if X is infinite), by elements of P (n)i(X) for i *
*> 0,
together with one element of P (n)0(X) for each component of X.
This result is essential for the calculations in Ravenel-Wilson-Yagita [RWY*
*98 ].
One version was stated as Theorem 1.11 of [Ya84 ], without proof (although the *
*ap-
proach suggested is now known not to work). In x8, our machinery of additive un*
*stable
operations provides a very short direct proof in terms of an explicit formula.
- 5 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
We also refine Landweber's filtration theorem. Yosimura [Yo76 , Thm. 3.4] *
*and
Yagita [Ya76 ] both observed that Landweber's theorem generalizes to stable P (*
*n)-co-
homology comodules M. The only finitely generated invariant prime ideals in P (*
*n)*
are Im = (vn, vn+1, . .,.vm-1 ) for n m < 1 (where In is interpreted as (0)).*
* We find
in x8 that an unstable comodule structure on M (in the sense of [BJW95 , Defn.*
* 6.32])
restricts the possible Landweber factors as follows.
Lemma 1.17 Let M be a P (n)*-module with a single generator x 2 Mk (in homol*
*ogy
degree -k) and annihilator ideal Ann (x) = Im , where n m < 1, so that
M ~= -kP (m)* ~= kP (m)* ~= k(P (n)*=Im ).
Then M admits an unstable P (n)-cohomology comodule structure if and only if k
satisfies the appropriate condition (depending on m and p):
(i)k 0 if m = n;
(ii)k g(n, n) - 1 if m = n + 1;
(iii)k g(n, m-1) - 1 if m n + 2 and p is odd;
(iv)k g(n, m-1) - 2 if m n + 2 and p = 2;
and this comodule structure is unique.
This leads directly to the filtration theorem.
Theorem 1.18 Let M be an unstable P (n)-cohomology comodule of finite type
(each Mi a finitely generated Fp-module) and bounded above (Mi = 0 for all i > *
*i0).
Then M admits a finite filtration by subcomodules
0 = M0 M1 . . .Mh = M
in which each quotient Mi=Mi-1is a monogenic comodule -kiP (mi)* with generator
xi, as listed in Lemma 1.17. In particular, M is a finitely presented P (n)*-mo*
*dule.
If, in addition, M is a P (n)*-algebra of any of the forms:
(i)M = P (n)*(X), for a finite complex X;
(ii)M = Im[f*: P (n)*(Y ) ! P (n)*(X)], for a map of spaces f: X ! Y , whe*
*re
X is a finite complex;
(iii)A spacelike (see [BJW95 , Defn. 7.14]) unstable P (n)*-cohomology al*
*gebra;
we may take each Mi to be an invariant ideal in M. At the last stage, we may ta*
*ke
xh = 1 and mh = n.
Our proof in x8 quotes the method of proof of Theorem 20.11 in [BJW95 ]. H*
*ow-
ever, here we prove that M is finitely presented, instead of assuming it. (Of c*
*ourse,
it has long been known that for finite X, P (n)*(X) is a coherent P (n)*-module*
* and
hence finitely presented.) In [ibid.], we overlooked the fact that this modifi*
*cation
applies equally well to BP = P (0), as follows. (Again, (i) is not new. However*
*, (ii)
is non-trivial and new when BP *(Y ) has phantom classes.)
Theorem 1.19 Let M be an unstable BP -cohomology comodule of finite type (ea*
*ch
Mi a finitely generated Z(p)-module) and bounded above, for example:
- 6 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
(i)M = BP *(X), for a finite complex X;
(ii)M = Im[f*: BP *(Y ) ! BP *(X)], for a map of spaces f: X ! Y , where X
is a finite complex.
_
Then M is a finitely presented BP*-module. |_|
Homological dimension Our starting point is the Conner-Floyd Theorem [CF66 ,
Thm. 10.1], that the map of ring spectra from the unitary Thom spectrum MU to
the K-theory spectrum K determined by the Todd genus induces for finite X an
isomorphism of cohomology theories
~= *
K* MU* MU*(X) --! K (X).
A far-reaching analogue is the result
E(n, m)* P(n)*P (n)*(X) ~=E(n, m)*(X), (1.20)
where E(n, m) = v-1mP (n, m). A key ingredient of such results is knowledge of*
* the
homological dimension of various (co)homology modules.
The case m = n of (1.20) is due to Morava [Mo85 ] as part of his structure *
*theorem,
and is quoted and reproved in [JW75 ], as well as by Yagita [Ya76 ]. The case *
*n = 0,
along with results on the homological dimension of BP*(X), was proved by Johnso*
*n-
Wilson [JW73 , Rk. 5.13] by means of the splitting theorem for BP in [Wi75 ]. *
*Shortly
afterwards, Landweber [La76 ] reproved this case by using cohomology operations
instead of the splitting, establishing his exact functor theorem in the process*
*; however,
he was unable to recover Corollary 4.4 of [JW73 ], which gave an upper bound o*
*n the
homological dimension of BP*(X). Later, Morava and Yagita [Ya77 , Thm. 3.11]
showed that P (n)*(X) is a BP *(BP )-module. Yagita and Yosimura [Yo76 ] both u*
*sed
this fact to generalize the exact functor theorem to P (n), which fully include*
*s (1.20),
and obtain homological dimension results for P (n)*(X).
We have now gone full circle, and with our splitting for P (n) in hand, can*
* use the
techniques of [JW73 ] to recover these results as well as (1.20), with the add*
*ed benefit
of the following estimate, which we establish in x5.
Theorem 1.21 Assume that X is a finite complex of dimension less than g(n, m*
*)=2.
Then the homological dimension of the P (n)*-module P (n)*(X) is at most m - n.
Although the exact functor theorem does not apply, ae(m): P (n) ! P (n, m) *
*still
induces a natural homomorphism of P (n, m)*-modules
____
ae(m): P (n, m)* P(n)*P (n)*(X) --! P (n, m)*(X).
This is an isomorphism when X is a point, but not in general, as the left side *
*is not
a cohomology theory. Classically, as in [JW73 ], one then asks for which X it *
*is an
isomorphism. Instead, we show in x5 that it is always an isomorphism in a certa*
*in
range of degrees [with no modification if p = 2]. Explicitly, its components are
____ OE X j
ae(m): P (n)h(X) vjP (n)h+2(p -1)(X) --! P (n, m)h(X). (1.22)
j>m
- 7 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Theorem 1.23 Assume that X is finite-dimensional and that m n > 0. Then
(1.22) is an isomorphism for all h g(n, m), and therefore a P (n, m)*-module *
*iso-
morphism in this range.
In particular, for m = n we have the isomorphism
____ OE X j
ae(n): P (n)h(X) vjP (n)h+2(p -1)(X) ~=k(n)h(X)
j>n
for all h 2pn, which preserves the vn-action in this range.
2 The ring spectrum P (n)
As the literature is somewhat conflicted [especially when p = 2], we review*
* the
construction of P (n) in fair detail. In this section, we work entirely in the*
* graded
stable homotopy category Stab*.
The spectrum P (n), so named by Johnson-Wilson [JW75 ], was based on work *
*of
Morava. It may conveniently be constructed directly from the Thom spectrum MU
by applying Sullivan-Baas theory [Ba73 ] to kill off the unwanted generators of*
* MU*,
as well as p (with no need for localization). (As stable P (n)-cohomology opera*
*tions
act faithfully on P (n)-homology, no information is lost by working in homology*
*.)
It is automatically a BP -module spectrum, with an action map ~: BP ^ P (n)*
* !
P (n) that satisfies the usual two module axioms, and the canonical map BP ! P *
*(n)
is BP -linear. It comes equipped with an exterior algebra E(Q0, Q1, . .,.Qn-1) *
*of BP -
linear operations, where Qi has homology degree -(2pi-1); we write the monomial
basis elements as QI = Qi00OQi11O. .O.Qin-1n-1for each multi-index I = (i0, i1,*
* . .,.in-1),
where each ir is 0 or 1.
The multiplication The canonical map j: S0 ! BP ! P (n) serves as the unit
map of P (n), where S0 denotes the sphere spectrum, but there is no obvious mul-
tiplication on P (n). It is known that for p 6= 2, there is a unique multiplic*
*ation
OE: P (n) ^ P (n) ! P (n) having the following properties:
(i)OE is BP -bilinear;
(ii)BP ! P (n) is multiplicative;
(iii)OE has j: S0 ! P (n) as two-sided unit;
(2.1)
(iv)OE is commutative;
(v) OE is associative;
(vi)Each Qi: P (n) ! P (n) is a derivation, in the sense that
QiOOE = OEO(Qi^ id) + OEO(id^ Qi): P (n) ^ P (n) --! P (n). (2.2)
Historically, three quite different approaches have been used. First, for *
*p 6= 2,
Morava [Mo79 ] used averaging over the symmetric group 2 to produce idempotent
operations in (co)bordism with repeated singularities. These operations yield a*
* canon-
ical multiplication OE on P (n) that is automatically commutative (cf. Mironov *
*[Mi78 ,
Thm. 4.2]). Associativity by this method involves averaging over 3 and requir*
*es
p 5 [ibid., Thm. 4.1].
- 8 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
The second method is heavily geometric. Mironov [Mi75 ] and Shimada-Yagita
[SY76 ] constructed (roughly equivalent) explicit multiplications on P (n) in t*
*he Baas
bordism context for any prime p. These apparently depend on a sequence of choic*
*es
of Morava manifolds. They automatically satisfy axioms (i), (ii) and (iii). Mor*
*eover,
Shimada-Yagita [SY76 , Thm. 5.25] and Mironov [Mi78 , Thm. 2.4] both show that
the obstructions to associativity lie in groups that vanish, and also obtain (v*
*i). The
disadvantage of this approach is that uniqueness is difficult to handle.
Third, W"urgler [W"u77 ] developed an entirely algebraic cohomological appr*
*oach
in terms of comodules, which leads to the existence of OE and the following res*
*ults.
Lemma 2.3 In the graded stable homotopy category Stab*:
(a) Any BP -linear map P (n) ! P (n), of any degree, can be uniquely written*
* in
the form X
cIQI: P (n) --! P (n), (2.4)
I
with coefficients cI 2 P (n)* of the appropriate degrees;
(b) Any BP -bilinear map P (n) ^ P (n) ! P (n), of any degree, can be unique*
*ly
written in the form
X
cI,JOEO(QI ^ QJ): P (n) ^ P (n) --! P (n), (2.5)
I,J
with coefficients cI,J2 P (n)* of the appropriate degrees;
(c) Any BP -trilinear map P (n) ^ P (n) ^ P (n) ! P (n), of any degree, can *
*be
uniquely written in the form
X
cI,J,KOEO(OE ^ id)O(QI ^ QJ ^ QK ): P (n) ^ P (n) ^ P (n) --! P (n),(2*
*.6)
I,J,K
with coefficients cI,J,K2 P (n)* of the appropriate degrees.
Proof Part (a) is a strengthened form of Proposition 3.5 of W"urgler [W"u77 ]*
*. Part
_
(b) is Proposition 4.12 of [ibid.], and (c) is entirely analogous. |_|
Lemma 2.7 The canonical map ae: P (n) ! P (n+1) is a map of ring spectra.
Proof By a slight generalization of (2.5) (also proved by W"urgler), any BP -*
*bilinear
map P (n) ^ P (n) ! P (n+1), in particular OEO(ae ^ ae), can be written
X
cI,JaeOOEO(QI ^ QJ): P (n) ^ P (n) --! P (n+1),
I,J
with coefficients cI,J2 P (n+1)*. Since OEO(ae ^ ae)O(j ^ j) = j, the sparsene*
*ss of
P (n+1)* leaves aeOOE as the only candidate for OEO(ae ^ ae). [This works even *
*for p = 2,
_
regardless of the choices of multiplication on P (n) and P (n+1).] |_|
If we write OEO(j ^ id) in the form (2.4), the sparseness of P (n)* yields *
*axiom (iii)
[even for p = 2], since we know OEO(j ^ j) = j. Then (ii) is a formal consequen*
*ce of
(i), (iii), and the BP -linearity of the map BP ! P (n).
- 9 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Since any BP -bilinear multiplication can be written in the form (2.5), the*
* sparse-
ness of P (n)* ensures that OE is unique, as long as p 3. Further, (iv) holds*
*, since
OEOT also satisfies (i) and (iii), where T : P (n) ^ P (n) ! P (n) ^ P (n) deno*
*tes the
switch map.
We may similarly deduce the associativity of OE, provided p 5, by writing*
* OEO(id^
OE) in the form (2.6). We also obtain (vi), provided p 3, by writing QiOOE i*
*n the
form (2.5); since (QiOOE)O(j ^ id) = Qi= (QiOOE)O(id^ j), the only candidate is*
* (2.2).
Finally, we should mention that there is now a fourth approach, the brave n*
*ew ring
context of Elmendorf-Kriz-Mandell-May. See [EKMM96 ] for p odd [or Strickland
[St99] for p = 2].
The case p = 2 It is well known that there is no commutative multiplication on
P (n) when p = 2. Instead, we see in [Bo ] that there are exactly two multiplic*
*ations
that satisfy all the axioms (2.1) except (iv). To make P (n) a ring spectrum, *
*we
arbitrarily_choose one of the two good multiplications as OE; then_the other is*
* its op-
posite, OE = OEOT , which defines the opposite ring spectrum P (n). Nassau [Na*
*02 ,
Thm. 3] shows_that_complex conjugation defines an isomorphism of ring spectra
: P (n) ~=P (n). _
Mironov [Mi78 , Thm. 4.7] computedOE explicitly in the form (2.5) as
_
OE = OEOT = OE + vnOEO(Qn-1 ^ Qn-1): P (n) ^ P (n) --! P (n). (2.8)
From now on, we write Q = Qn-1, in view of its frequent occurrence.
Products in homology and cohomology We review briefly the various products
in P (n)-(co)homology. Their properties are familiar enough [except when p = 2]*
*. We
remind the reader that the operations Qi act on both homology and cohomology.
Given x 2 P (n)*(X) and y 2 P (n)*(Y ), we have the cohomology cross product
x x y 2 P (n)*(XxY ); by taking Y = X and using the diagonal map of X, we deduce
the cup product xy 2 P (n)*(X), which makes P (n)*(X) a ring. Given a 2 P (n)*(*
*X)
and b 2 P (n)*(Y ), we have the homology cross product a x b 2 P (n)*(X xY ). *
*All
three products are associative. For p 6= 2, they are also commutative, in the *
*sense
that T *(yxx) = x x y, yx = xy, and T*(bxa) = a x b. By equation (2.2), ea*
*ch
Qi is a derivation for all three products.
By taking X as a one-point space, P (n)*(Y ) and P (n)*(Y ) become P (n)*-m*
*odules,
and both cross products are P (n)*-bilinear [even for p = 2; see below].
There is also the scalar or Kronecker product 2 P (n)* of x 2 P (n)**
*(X) and
a 2 P (n)*(X), which is P (n)*-bilinear [even for p = 2; see [Bo ]].
The case p = 2 There are of course no signs, but the noncommutativity of OE fo*
*rces
us to watch carefully for any shuffling of copies of P (n). Nevertheless, we fi*
*nd [Bo ]
that the K"unneth and duality formulae continue to hold, exactly as stated in [*
*Bo95 ].
It is immediate from equation (2.8) that
__ *
T *(yxx) = x x y = xxy + vnQxxQy in P (n) (X xY ), (2.9)
__
where_x x y denotes the twisted cross product formed using the opposite multipl*
*ica-
tionOE on P (n). For cup products, this implies
yx = xy + vn(Qx)(Qy) in P (n)*(X), (2.10)
- 10 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
so that P (n)*(X) is not commutative in general in the ordinary sense. Alterna-
tively, these products are TQ -commutative if we replace the standard commutati*
*vity
isomorphism T : A B ~=B A everywhere by TQ : A B ~=B A, defined by
TQ (a b) = b a + vnQb Qa in B A. (2.11)
Similarly, homology is also TQ -commutative, in the sense that
__
T*(bxa) = a x b = a x b + vnQa x Qb in P (n)*(X xY ). (2.12)
Taking X to be a point shows that the P (n)*-actions on P (n)*(Y ) and P (n*
*)*(Y )
are independent of the choice of OE. In [Bo ], we find that is also inde*
*pendent of
this choice.
There is one surprise, on account of the hidden shuffling, proved in [Bo ].
Proposition 2.13 Given x 2 P (n)*(X), y 2 P (n)*(Y ), a 2 P (n)*(X), and b 2
P (n)*(Y ), we have
= + vn. (2.14)
If instead we mix the products, we find
__
= . (2.15)
3 Proofs of the main theorems
In this section, we establish Theorems 1.2 and 1.12. More precisely, we re*
*duce
them to two key lemmas: Lemma 3.1 provides our main splitting and Lemma 3.8 will
imply that our splittings are best possible.
Splittings All our splittings are derived from the following splitting.
Lemma 3.1 For k g(n, m), where m n, there is a map
____
`(m) : P_(n,_m)_k--!P_(n)_k
____
that splits the canonical map ae(m): P_(n)_k! P_(n,_m)_k, i. e. ae(m)O`(m) ' id*
*. It is a
map of H-spaces [except when p = 2 and k = g(n, m)].
We express this in terms of idempotent P (n)-cohomology operations in x5.
A short direct proof of Lemma 3.1 is presentediin [BW01j ], based on theiba*
*r j
spectral sequence. For such k, we show that E* P_(n,_m)_kis a quotient of E* P_*
*(n)_k,
first for E = P (n), then_for_E = P (n, m), and that these are free E*-modules.*
* It
follows by duality that`(m) exists, but its status as an H-map is left unclear.
We deduce other useful splittings. The canonical map ae(m-1, m): P (n, m) !
P (n, m-1), which kills vm , fits into the exact triangle of spectra
P (n, m) -vm--!P (n, m) -ae(m-1,m)------!P (n, m-1) -ffi-!P (n,(m).*
*3.2)
On homotopy groups, this induces the obvious short exact sequence
0 ! Fp[vn, vn+1, . .,.vm ] -vm--!Fp[vn, vn+1, . .,.vm ] ! Fp[vn, vn+1, . .,.*
*vm-1 ] ! 0.
- 11 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Unstably, we have the H-space fibration
P_(n,_m)_k+2(pm -1)vm---!P_(n,_m)kae(m-1,m)-------!P_(n,km-1).
For k g(n, m-1), the composite
_____
P_(n,_m-1)_k-`(m-1)----!P_(n)kae(m)----!P_(n,km) (3.3)
automatically splits ae(m-1, m), to yield the decomposition
P_(n,_m)_k' P_(n,_m-1)_kx P_(n,_m)_k+2(pm,-1) (3.4)
where the two injections are (3.3) and vm . This is a decomposition of H-spaces*
* [except
when p = 2 and k = g(n, m-1)].
Proof of Theorem 1.12 This is completely analogous to the proof of Theorem 1.16
of [BJW95 ]. Everything we need is contained in the commutative diagram
P_(n)_k+2(pj-1)______P_(n)_k-vj______-P_(n,_m)_kae(m)
| | j3
| | j
| | j j
|ae(j) |ae(j) j (3.5)
| | j ae(m,j)
|? |? j
P_(n,_j)_k+2(pj-1)___P_(n,_j)_k-vj
of H-spaces and canonical H-maps, where j > m.
With m given by (1.11), we observe that the spaces Yk and Yk+2(pj-1)appear
in the diagram disguised as P_(n,_m)_kand P_(n,_j)_k+2(pj-1). We insert the spl*
*ittings
____ ___
`(m) and`(j) from Lemma 3.1 to produce the desired decomposition of P_(n)_k, *
*as
suggested by the decomposition of abelian groups
M
Fp[vn, vn+1, vn+2, . .].= Fp[vn, vn+1, . .,.vm ] vjFp[vn, vn+1, . .*
*,.vj].
j>m
(But we warn that our splittings cannot be expected to induce exactly this deco*
*m-
position of the coefficient ring P (n)*,_and_it seems likely that they never do*
*.)
In detail, we map Yk into P_(n)_kby`(m) , which is an H-map [unless p = 2 a*
*nd
k = g(n, m)], and Yk+2(pj-1), for each j > m, by the H-map (in all cases)
___ v
Yk+2(pj-1)`(j)---!P_(n)k+2(pj-1)j--!P_(n)k.
We multiply these together, using the H-space structure of P_(n)_k, to form a m*
*ap
f: W ! P_(n)_kfrom the restricted direct product W (the union of all finite sub*
*prod-
ucts) of the based Y-spaces mentioned. ___
We filter P (n)* by the ideals Jj. We note that vjO`(j)induces a homomorphi*
*sm
P (n, j)* ! Jj-1 on homotopy groups that induces the quotient isomorphism
vj
P (n, j)* = Fp[vn, vn+1, . .].=Jj --! Jj-1=Jj.
This is enough to guarantee that f induces an isomorphism on homotopy groups and
is thus a homotopy equivalence. Because the connectivities of the Y-spaces incr*
*ease,
W is homotopy-equivalent to the desired full product and we have (1.13).
- 12 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
The same method applies to P_(n,_h)_k, with the simplification that the pro*
*duct
W is now finite. (One can also produce decompositions like (1.14) directly from*
* the
splittings (3.4) by induction on h, though the resulting maps are different and*
* far
more complicated.) _______
For k = 0, the splitting`(n-1) : Y0 = Fp ! P_(n)_0is obvious and unique up*
* to
_
homotopy. We can still use diagram (3.5). |_|
Indecomposability On the other hand, we need to know that Yk does not split.
Lemma 3.6 A map f: Yk ! Yk is a homotopy equivalence if and only if it induc*
*es
an isomorphism on the bottom homotopy group ssk(Yk) ~=Fp.
_
Corollary 3.7 The space Yk does not decompose as a product. |_|
In x12, we prove the following about P (n) and deduce Lemma 3.6 from it.
Lemma 3.8 Represent an unstable operation r: P (n)k(-) ! P (n)m (-), where k*
* > 0
and m > 0, by the map r: P_(n)_k! P_(n)_m. Then the induced homomorphism on
homotopy groups
i j ss (r) i j
r*: kP (n)* ~=ss* P_(n)_k- --*-!ss* P_(n)_m ~= m P (n)* (3.9)
has the properties, for any element v 2 P (n)*:
(a) r* k(vnv) = vnr* kv;
(b) r* k(vqv) vqr* kv mod Iq = (vn, vn+1, . .,.vq-1), provided k > g(n, q-*
*1).
Construction of maps Our strategy for proving Theorem 1.2 is to construct
enough maps to and from the spaces P_(n)_k.
Lemma 3.10 If X is a space for which k(n)*(X) is a free k(n)*-module, then
P (n)*(X) is a free P (n)*-module.
Proof Lemmas 4.7 (with k = m = n) and 2.1 of Yosimura [Yo76 ] show that P (n)*
**(X)
_
is a flat P (n)*-module. Such modules are free by [ibid., Prop. 1.5]. |_|
Lemma 3.11 Let X be a (k-1)-connected space with ssk(X) a nonzero finite abe*
*lian
p-group and suppose k(n)*(X) is a free k(n)*-module. Then there existsiajmap
f: X ! P_(n)_kthat induces a nonzero homomorphism f*: ssk(X) ! ssk P_(n)_k ~=Fp
on the bottom homotopy groups.
Proof Since P (n)*(X) is a free P (n)*-module by Lemma 3.10, the universal co*
*effi-
cient theorem [Bo95 , Thm. 4.14] gives
P (n)*(X) ~=Hom *P(n)*(P (n)*(X), P (n)*).
As X is (k-1)-connected, P (n)k(X) ~=Hk(X; Fp) ~=ssk(X) Fp 6= 0, and it is cl*
*ear
*
* _
that suitable cohomology classes f 2 P (n)k(X), i. e. maps f: X ! P_(n)_k, exis*
*t. |_|
Proof of parts (a) and (b) of Theorem 1.2 We first note that for k > 0, the sp*
*ace
P_(n)_ksatisfies the axioms (1.1). Axiom (i) is clear. Axiom (ii) holds by [R*
*W96 ].
- 13 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
i j
Axiom (iii) is easy. Take any element kv 2 kP (n)* ~=ss* P_(n)_k, where v 2 P*
* (n)h.
Viewed as a cohomology class, it is vuk+h 2 P (n)*(Sk+h). Multiplication by v *
*on
P (n)*(-) is represented by the map we want, v: P_(n)_k+h! P_(n)_k.
Then Yk, being a retract of P_(n)_k, also satisfies the axioms. By Corolla*
*ry 3.7,
it is indecomposable. Uniqueness of Yk and our decompositions will follow from *
*(b),
under the assumption that all our H-spaces have finite type.
For the induction step in (b), given any (k-1)-connected space X that satis*
*fies
the axioms, define m by (1.11). Then Lemma 3.11 provides a map
h: X --! P_(n)_k-ae(m)---!P_(n,_m)k= Yk
that induces a nonzero homomorphism h*: ssk(X) ! ssk(Yk) ~=Fp. Choose ff 2 ssk(*
*X)
such that h*ff = 1 2 Fp; then axiom (iii) provides a map
___
f: Yk = P_(n,_m)_k`(m)----!P_(n)k--!X
that induces f*1 = ff. By Lemma 3.6, hOf: Yk ! Yk is a homotopy equivalence. We
use the homotopy fibre j: F ! X of h and the multiplication ~ on X to construct*
* a
homotopy equivalence
Yk x F --fxj-!X x X -~-!X.
Then F , being a retract of X, again satisfies the axioms.
We begin the induction with Z0 as the given space, and find a sequence of e*
*quiv-
alences Zi' Ykix Zi+1for i 0. By finiteness, the spacesQZibecome more and more
_
highly connected as i increases, and we deduce Z0 ' iYkias required. |_|
4 k(n)-towers with vn-free homotopy
In this section, we prove Theorem 1.6. We must show that the original axiom*
* (iii)
of (1.1) is equivalent (in the presence of the other axioms) to axiom (iii)0 st*
*ated in
(1.7), which asserts that X is a k(n)-tower with vn-free homotopy. Lemma 4.1 sh*
*ows
that (iii)0implies (iii), while Lemma 4.3 gives the converse.
Lemma 4.1 Suppose the connected H-space X is a k(n)-tower of finite type with
vn-free homotopy. Then axiom (iii) holds: given k > 0, any map Sk ! X factors
through the standard map uk: Sk ! P_(n)_kto yield a map P_(n)_k! X.
We first show that it does not matter how far up the tower we can lift.
Lemma 4.2 In diagram (1.4), any map f: P_(n)_k! Xi-1lifts to a map P_(n)_k! *
*X.
Proofi Withjzi as in Definition 1.3, weinotejthat vcn(ziOf) =i f*(vcnzi)j = *
*0 in
k(n)* P_(n)_k. But by [RW96 ], k(n)* P_(n)_k and hence k(n)* P_(n)_k contain *
*no
vn-torsion; therefore ziOf ' 0 and f lifts to f0: P_(n)_k! Xi.
_
By induction and limits, f lifts all the way to X. |_|
Proof of Lemma 4.1 For any connected space Y and k > 0, let us call an element
ff 2 ssk(Y ), or map ff: Sk ! Y , extendable if it extends over uk to a map P_(*
*n)_k! Y .
- 14 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
i j
All elements of ss* k(n)_q ~= qFp[vn] are obviously extendable. It follow*
*s from
diagram (1.5) that every element in Ker[ss*(Xi) ! ss*(Xi-1)] is extendable.
By Lemma 4.2, any extendable element of ss*(Xi-1) lifts in diagram (1.4) to*
* some
extendable element of ss*(X).
The sum of any two extendable elements of ssk(X) is again extendable: given
f1, f2: P_(n)_k! X, we use the given multiplication ~ on X to construct the map
P_(n)_k--!P_(n)_kx P_(n)_kf1xf2----!X x X -~-!X.
_
Together, these facts imply that every element of ss*(X) is extendable. |_|
The space Yk A countable product of k(n)-towers with vn-free homotopy is anoth*
*er
such tower (provided it has finite type). In view of Theorem 1.2(b), it suffice*
*s to prove
the following.
Lemma 4.3 For each k > 0, the space Yk is a k(n)-tower with vn-free homotopy.
We first destabilize the Johnson-Wilson construction [JW75 , x4] of a filt*
*ration
of the spectrum P (n) whose subquotients are suspensions of k(n), and adapt it *
*for
P (n, m). The result will be a tower
. .-.-!W3 --! W2 --! W1 --! W0 = P_(n,_m)_k (4.4)
with trivial homotopy limit, where each Wi is the homotopy fibre of a map Wi-1!
k(n)_q(i)that is epic on homotopy groups. This depends on the following lemma,
i j
where we recall that ss* P_(n,_m)_k~= kP (n, m)* etc.
Lemma 4.5 Given v 2 P (n, m)h and k g(n, m), there exist an integer c and *
*stable
P (n)-operation r such that the composite
___
s: P_(n,_m)_k`(m)----!P_(n)kr--!P_(n)k+h-2cNae(n)---!k(n)_k+h-2cN(4.6)
induces s* kv = k+h-2cNvcnon homotopy groups.
_
Proof Lemma 1.12 of [JW75 ], viewed unstably, supplies c and r. |_|
We construct the tower (4.4) by induction, starting from W0 = P_(n,_m)_k.
Given j: Wi-1 ! P_(n,_m)_k, where Wi-1 is (k+h-1)-connected and j*: ss*(Wi-1) !
i j
ss* P_(n,_m)_kis monic, we choose a bottom nonzero element u 2 ssk+h(Wi-1) to k*
*ill.
Lemma 4.5 provides a map s: P_(n,_m)_k! k(n)_k+h-2cNsuch that s*j*u = k+h-2cNv*
*cn.
For dimensional reasons, sOj factors through vcn: k(n)_k+h! k(n)_k+h-2cNto prod*
*uce
the desired map Wi-1! k(n)_k+h, with fibre Wi.
This is the wrong kind of tower for Definition 1.3. To correct it, we could*
* take the
homotopy fibre Xiof each map Wi! P_(n)_k, to express P_(n,_m)_k= P_(n,_m)_k-1as
a k(n)-tower with vn-free homotopy. This approach fails to produce a suitable t*
*ower
for P_(n,_m)_kwhen k = g(n, m). Our solution is to observe that it is ineffici*
*ent to
deloop and then take fibres; instead, we prove only what we actually need.
- 15 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Lemma 4.7 Given a (k+h)-connected map q: P_(n,_m)_k! X and any map
s: P_(n,_m)_k! k(n)_k+h-2cN, there exists a vn-torsion map z: X ! k(n)_k+h+1such
that s is one value of the following Toda bracket,
s 2 : P_(n,_m)_k--!k(n)_k+h-2cN.
Proof We are using the adjoint (but equivalent) description of a Toda bracket*
* in
terms of loop spaces instead of suspensions. We build the commutative diagram
Figure 4.1 in which the two rows are fibration sequences. We start with the obv*
*ious
P_(n,_m)_k
| Q
| Q q
|q0 Q
| Q
|? QQs
k(n)_k+h _________X0-____________X- ________-k(n)_k+h+1z
| |
| | | |
|= |g |f |=
| | | |
|? |? |? |?
-vcn ss ffi vcn
k(n)_k+h _____k(n)_k+h-2cN-___G_k+h-2cN-_____k(n)_k+h+1-__k(n)_k+h-2cN+1-
Figure 4.1: Construction of the Toda bracket
fibration as the bottow row, where (stably) G denotes the cofibre ofivcn: k(n)j*
*! k(n),
with homotopy Fp[vn]=(vcn). By the connectivity of q, q*: Gj(X) ! Gj P_(n,_m)_k*
*is
an isomorphism for j k + h - 2cN + 2N - 1, so that ssOs factors uniquely thro*
*ugh q
to yield a map f such that f Oq = ssOs. We put z = ffiOf, which automatically s*
*atisfies
vcnz = 0. We define X0 as the homotopy fibre of z, and fill in g to form a morp*
*hism
of fibrations.
Since X0 may be constructedhas a pullback, weican fill in q0 to lift q and *
*satisfy
gOq0 = s. (Equivalently, vcnOP_(n,_m)_k, k(n)_k+his part of the indeterminacy o*
*f the
_
Toda bracket.) Then by definition, gOq0= s is one value of the Toda bracket. |*
*_|
Proof of Lemma 4.3 We build the desired tower for P_(n,_m)_kby induction, star*
*ting
from a point as X0. Suppose we have constructed a map qi-1: P_(n,_m)_k! Xi-1
that induces a surjection qi-1*: kP (n, m)* ! ss*(Xi-1) on homotopy groups, wi*
*th
kernel K an Fp[vn]-submodule. We choose a bottom nonzero element kv 2 Kk+h to
kill, where Ki = 0 for i < k + h. Then Lemma 4.5 provides a map s: P_(n,_m)_k!
k(n)_k+h-2cN. We use Lemma 4.7 to build Figure 4.1, taking qi-ias q and Xi-1as *
*X.
_We_next take homotopy groups of Figure 4.1. By Lemma 3.8(a), applied to
rO`(m) Oae(m): P_(n)_k! P_(n)_k+h-2cN, s* is a homomorphism of Fp[vn]-modules. *
*By
exactnessiandjthe hypothesis that q* k(vinv) = 0, q0*(vinv) must lift to - k+hv*
*in2
ss* k(n)_k+h. It now follows that q0*is also epic, with kernel
h i
K0= Ker s*|K: K --! k+h-2cNFp[vn] K,
a strictly smaller Fp[vn]-submodule of kP (n, m)*. We take X0 as Xi and q0 as *
*qi.
- 16 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
The kernels K become more and more highly connected as i increases, hence
_
P_(n,_m)_kis the homotopy limit of the spaces Xi. |_|
5 Splittings of P (n)-cohomology
In this section, we translate the H-space splittings in x3 into splittings *
*of P (n)-
cohomology. We also deduce Theorems 1.21 and 1.23.
We have yet to prove Lemmas 3.1, 3.6 and 3.8. Lemma 3.1 is equivalent to the
following statement for the represented functors. (We do not mention Lemmas 3.6
and 3.8 again until x12.)
Lemma 5.1 Assume that k g(n, m), where m n. Then there is a splitting
____
`(m) : P (n, m)k(X) --! P (n)k(X)
____
of ae(m): P (n)k(X) ! P (n, m)k(X) that satisfies ae(m)O`(m) = idand is natural*
* for
spaces X. It is additive [except when p = 2 and k = g(n, m)].
This we actually prove in x9 [except the nonadditive case; see [Bo ]], by c*
*onstructing
an idempotent cohomology operation `(m) in P (n)k(X). Unlike the case of BP , t*
*he
use of nonadditive operations yields no further splittings [unless p = 2].
We next translate equation (3.4).
Corollary 5.2 For k g(n, m-1), where m > n, we have the natural short exact
sequence of abelian groups
m -1) vm k ae(m-1,m) k
0 ! P (n, m)k+2(p (X) ---! P (n, m) (X) -------! P (n, m-1) (X) ! 0.
_
This splits naturally [unless p = 2 and k = g(n, m-1)]. |_|
This implies our homological dimension bound, by the methods of [JW73 ].
Proof of Theorem 1.21 Following Yosimura [Yo76 , Thm. 4.8], we need to show th*
*at
ae(m-1, m): P (n, m)i(X) --! P (n, m-1)i(X) (5.3)
is epic for all i. For i 2(pm -1), this is trivial, by the exact sequence
P (n, m)i(X) -ae(m-1,m)------!P (n, m-1)i(X) -ffi-!P (n, m)i-2(pm -1)-1*
*(X)
arising from the exact triangle (3.2).
For i > 2(pm -1), we embed X in R2q+1, where q is the dimension of X, and t*
*ake
a regular neighborhood V of X. By Poincar'e duality, (5.3) is equivalent to
ae(m-1, m): P (n, m)2q+1-i(V, @V ) --! P (n, m-1)2q+1-i(V, @V ).
This is epic by Corollary 5.2, because by hypothesis
_
2q + 1 - i (g(n, m) - 2) + 1 - (2(pm -1) + 1) = g(n, m-1). |_|
We also translate Theorem 1.12, using the splittings made explicit in x3, a*
*nd
finally deduce Theorem 1.23. (Decompositions like (5.6) also follow directly f*
*rom
Corollary 5.2 by induction on h, though the resulting homomorphisms are differe*
*nt.)
- 17 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Theorem 5.4 Let X be any space and suppose that m n > 0.
(a) If k g(n, m) [replaced by k < g(n, m) if p = 2], we have the natural a*
*belian
group decomposition
Y j
P (n)k(X) ~=P (n, m)k(X) P (n, j)k+2(p -1)(X), (5.5)
j>m
____
where the first factor on the right is injected by`(m) , and the others by
j-1) `___(j) k+2(pj-1) vj k
P (n, j)k+2(p (X) ---! P (n) (X) --! P (n) (X).
Hence, by composition with ae(h): P (n)k(X) ! P (n, h)k(X) for any h > m,
Mh j
P (n, h)k(X) ~=P (n, m)k(X) P (n, j)k+2(p -1)(X). (5.6)
j=m+1
These decompositions are maximal if k > g(n, m-1) [also for k = g(n, m-1) if p *
*= 2].
(They are in no sense decompositions as P (n)*-modules.)
(b) If p = 2 and k = g(n, m), we replace equations (5.5) and (5.6) by the na*
*tural
short exact sequences
Y j+1 ae(m)
0 ! P (n, j)k+2 -2(X) ! P (n)k(X) ----! P (n, m)k(X) ! 0
j>m
and
Mh j+1 ae(m,h) _
0 ! P (n, j)k+2 -2(X) ! P (n, h)k(X) -----! P (n, m)k(X) ! 0. |_|
j=m+1
Because P (n, n) = k(n) is so familiar, we break out the special case m = n*
*. For
h < 2(pn-1), we can even replace k(n) by the periodic Morava K-theory K(n).
Corollary 5.7 For h 2pn, where n > 0, we have, for all spaces X, the natur*
*al
abelian group decomposition
Y j
P (n)h(X) ~=k(n)h(X) P (n, j)h+2(p -1)(X),
j>n
except that if p = 2 and h = 2n+1, we have only the natural short exact sequence
Y j+1 ae(n) _
0 ! P (n, j)h+2 -2(X) ! P (n)h(X) ---! k(n)h(X) ! 0. |_|
j>n
Remark All the splittings exhibited above depend on the choice of `(m), which*
* is
not canonical and does not respect multiplication by vj.
Proof of Theorem 1.23 As X is finite-dimensional,_the sum in equation (1.22) is
essentially_finite._Lemma 5.1 shows thatae(m) is epic. It is clear from Theorem*
* 5.4
_
that Kerae(m) is contained in the sum, and must therefore be the sum. |_|
- 18 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
6 Stable operations in P (n)-cohomology
In this section, we describe the stable operations in P (n)-cohomology P (n*
*)*(-)
in the style of [Bo95 ]. The results are old and well known [except for p = 2],*
* but we
include them for completeness and ease of reference; more importantly, they ser*
*ve as
a pattern for xx7, 10.
Monoidal structure (For the language of monoidal categories and functors, see
e. g. Mac Lane [Ma71 , Ch. VII].) Since P (n)* is a commutative ring [even if p*
* = 2],
the graded category (FMod *, b, P (n)*) of complete Hausdorff filtered P (n)*-*
*modules
is a symmetric monoidal category, with all (completed) tensor products taken ov*
*er
P (n)*. The cross product makes P (n)-cohomology a monoidal functor,
P (n)*(-): (Ho op, x, point) --! (FMod , b, P (n)*). (6.1)
(Conveniently, P (n)*(X) has no phantom classes and so is already complete Haus-
dorff.) For homology, we similarly have the monoidal functor
P (n)*(-): (Ho , x, point) --! (Mod , , P (n)*), (6.2)
with values in the category Mod of discrete P (n)*-modules. Both functors are *
*sym-
metric for p 6= 2.
The cohomology version for spectra and graded maps is
P (n)*(-, o): (Stabop*, ^, S0) --! (FMod *, b, P (n)*),
and similarly for homology. (We include the basepoint subspectrum o in our nota*
*tion
as a reminder that all stable (co)homology is reduced, and to distinguish it fr*
*om the
(co)homology of a space, which here will generally be absolute.)
Operations Because = P (n)*(P (n), o) is a free P (n)*-module, we may identi*
*fy its
dual P (n)*-module D with A = P (n)*(P (n), o), the algebra of all stable oper*
*ations
in P (n)-cohomology, and have available all the stable machinery and results of*
* [Bo95 ].
In particular, we have the monoidal functor
S: (FMod *, b, P (n)*) --! (FMod *, b, P (n)*) (6.3)
defined by SM = FMod *(A, M). If M is filtered by submodules F aM, we filter SM
by the submodules F aSM = SF aM; as in [ibid.], SM is again complete Hausdorff.
The ring spectrum structure of P (n) gives S its monoidal structure (see diagra*
*m (6.10)
below), which is symmetric for p 6= 2. (As in [Bo95 ], care is needed in keepin*
*g track
of the many P (n)*-module actions, some of which are not obvious.)
The action of stable P (n)-cohomology operations is visibly encoded*
* in the
monoidal natural transformation
aeX : P (n)*(X) --! S(P (n)*(X)) = FMod *(P (n)*(P (n), o), P (n)*(X))(6.4)
defined by aeX x = x*, where we treat x 2 P (n)*(X) as a map of spectra x: X+ !*
* P (n)
and X+ denotes the disjoint union of X and a (new) basepoint.
The coaction To convert the action of A into a coaction by , we recall the na*
*tural
isomorphism [Bo95 , (11.4)]
`M: S0M = M b ~=FMod *(D , M) ~=FMod *(A, M) = SM,
- 19 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
given on x 2 M, c 2 , and r 2 A ~=D by
((`M)(x c))r = x, (6.5)
with the expected sign. We use it to transfer all the structure from the functo*
*r S to
S0 and replace (6.4) by the equivalent natural transformation
aeX : P (n)*(X) --! S0P (n)*(X) = P (n)*(X) b . (6.6)
The monoid The resulting monoidal structure on S0 is necessarily induced by a
monoid structure on the P (n)*-module (as we see by naturality from the case
M = N = P (n)* in diagram (6.10), below), and conversely. We simply need to
compute it.
Lemma 6.7 The following monoid structure on , which is inherited from the
monoidal functor S, makes the natural transformation (6.6) monoidal:
(a) If p is odd, the multiplication on is the obvious one,
= P (n)*(P (n), o) P (n)*(P (n), o) -x-!P (n)*(P (n) ^ P (n), o)
- OE*-!P (n) (6.8)
*(P (n), o) = ,
as inferred by writing = P (n)*(P (n), o). The unit homomorphism of is
P (n)* = P (n)*(S0, o) -j*-!P (n)*(P (n), o) = .
(b) If p = 2, the multiplication is instead
__
= P (n)*(P (n), o) P (n)*(P (n), o) -x-!P (n)*(P (n) ^ P (n), o)
- OE*-!P (n) (6.9)
*(P (n), o) = ,
____
which is better suggested by writing =P (n)*(P (n), o). The unit is unaffecte*
*d.
Proof The multiplication OE on P (n) induces
OE*: D ~=P (n)*(P (n), o) --! P (n)*(P (n) ^ P (n), o) ~=D b D ,
with the help of the K"unneth formula [Bo95 , Thm. 4.19]. The natural transform*
*ations
i(M, N) for S0 and S form the left and right sides of the commutative diagram
`M `N
(M b ) b(N b ) ____________FMod- *(D , M) b FMod *(D , N)
| |
|~ |~
|= |=
|? |?
M b N b( ) FMod *(D b D , M b N) (6.10)
| |
|M N OE |FMod*(OE*,id)
| |
|? `(M bN) |?
M b N b _____________________-FMod *(D , M b N)
which features the multiplication OE: ! . We evaluate on x c y d,
where x 2 M, y 2 N, and c, d 2 . By (6.5), the lower route gives the element r*
* 7!
- 20 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
x y of FMod *(D , M b N). The upper route gives r s 7! ~~x y
in FMod *(D b D , M b N). Assuming p 6= 2, we can rewrite this as x
y; then in FMod *(D , M b N) we find
r 7! x y = x y.
Thus cd = OE*(cxd) (with no sign) as expected, which is (a).
[If p = 2, this calculation is_false;_we must use equation_(2.15) instead, *
*which
states that ~~~~ = . Then cd = OE*(cx d), for (b).]
The unit z: P (n)* ! SP (n)* takes 1 = j 2 P (n)* to the homomorphism
* * 0
D ~=P (n)*(P (n), o) -j-! P (n) (S , o) ~=P (n)*,
in other words, r 7! = . Comparison with (6.5) shows that the *
*corre-
_
sponding element of S0P (n)* = P (n)* ~= is j*1. |_|
If X is a point in (6.6), we find the right unit ring homomorphism
jR: P (n)* --! S0P (n)* = P (n)* ~= , (6.11)
which is used to make a right P (n)*-module (hence a bimodule). Since ae is
monoidal, this action makes (6.6) a homomorphism of P (n)*-modules.
The Hopf algebroid Now we add the algebra structure of A. Exactly as in [Bo95 ,
x10], composition of operations and the identity operation induce natural trans*
*forma-
tions _: S ! SS and ffl: S ! I. These make S a monoidal comonad in the category
FMod , and (6.4) makes P (n)*(X) an S-coalgebra.
We transfer this structure too to S0. The resulting monoidal comonad struct*
*ure on
S0is necessarily induced by a Hopf algebroid structure on (as we see by takin*
*g M =
P (n)*), and conversely. This structure consists of a coassociative comultipli*
*cation
_S: ! with counit fflS: ! P (n)*. These behave exactly as in Adams
[Ad74 ] or [Bo95 , Thm. 11.35]; in particular, _S and fflS are homomorphisms of*
* P (n)*-
bimodules and algebras. [This all works without change for p = 2; see [Bo ].]
Proposition 6.12 The stable operations in P (n)-cohomology_are encoded in the
*
* _
Hopf algebroid = P (n)*(P (n), o) [replaced by =P (n)*(P (n), o) for p = 2]*
*. |_|
The discussion of the structure of carries over from the case K(n) in [Bo*
*95 ] with
little change [except that we allow p = 2]. We even use the same test spaces.
The one-point space We already discussed this in (6.11). The coaction ae reduc*
*es
to the ring homomorphism jR, which is determined by the elements
wk = jRvk 2 2(pk-1) for k n. (6.13)
Complex orientation Our next test space is complex projective space C P 1. As
P (n) inherits a complex orientation from BP (or MU), we have P (n)*(C P 1) =
P (n)*[[x]], the formal power series ring generated by the Chern class x = x(,)*
* of the
Hopf line bundle , over CP 1, filtered by powers of the ideal (x).
The coaction ae for CP 1 defines elements bj 2 2j-2by the formula [Bo95 , *
*(13.2)]
1X
aex = b(x) = xj bj in P (n)*(C P 1) b ~= [[x]]. (6.14)
j=1
- 21 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Here, b(x) is a useful formal abbreviation for the right side. As always in the*
* stable
context [ibid., Prop. 13.4], b1 = 1 and b0 = 0.
Further, the comultiplication _S is given on bi as the coefficient of xi in
1X
_Sb(x) = b(x)j bj in ( )[[x]], (6.15)
j=1
and fflSbj = 0 for all j > 1.
Since P (n) is p-local, we need only the accelerated elements b(j)= bpj2 2*
*(pj-1)
for j 0, where b(0)= 1; the other b's are expressible in terms of these and t*
*he v's
and w's by [ibid., Lemma 13.7].
The p-th power map i: CP 1 ! CP 1, whose bundle interpretation is i*, = , p,
induces in cohomology
X1
i*x = [p](x) = gixi+1 in P (n)*(C P 1) = P (n)*[[x]]
i=N
for certain coefficients gi2 P (n)2i. This formal power series is known as the *
*p-series
for P (n). There are no lower terms as g0 = p = 0 in P (n)0. (The elements gi*
* are
traditionally written ai, but we rename them in order to avoid confusion with o*
*ther
elements, also named ai, that appear shortly.)
We need only one standard fact [RW77 , Thm. 3.11(b)] about the p-series:
k
[p](x) vkxp mod (vn, . .,.cvk, . .). (6.16)
for any k n, where the ideal is generated by all the v's except vk. In words,*
* [p](x)
k i q
contains terms vkxp but not ~vkx for any i > 1. In particular,
n pn+1
[p](x) = vnxp + vn+1x + higher terms. (6.17)
Hence as k varies, we have
X1 k
[p](x) vkxp mod V 2, (6.18)
k=n
where V denotes the maximal ideal (vn, vn+1, vn+2, . .). P (n)*.
Naturality of ae with respect to the map i yields the identity [Bo95 , (13.*
*11)]
X1
b([p](x)) = [p]R(b(x)) = b(x)i+1jRgi in [[x]]. (6.19)
i=N
n
The lowest power of x that occurs is xp .
Definition 6.20 For each k n, we define the k-th main stable relation (Rk) *
*as
k
the coefficient of xp in equation (6.19).
Since b(0)= 1, the first relation (Rn) is simply vn1 = wn, which implies th*
*at every
stable operation is vn-linear. For k > n, equation (6.18) shows that (6.19) has*
* a term
k
wkxp on the right, and (Rk) becomes an inductive formula for wk in terms of th*
*e v's
and b's and lower w's.
- 22 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Cohomology of a lens space, for p odd Our final test space is the 2N-skeleton
L of the lens space K(Z =p, 1). Geometrically, L is the orbit space of the sta*
*ndard
Z =p -action on the unit sphere S2N+1 C N+1 given as complex multiplication by
Z =p S1 C, with the top cell omitted by requiring the last coordinate to be*
* real
non-negative, up to the action of Z =p. (Retaining the top cell, as in [Bo95 ]*
*, adds
some extra complication but offers little benefit.)
Following [Bo95 , x14], its cohomology is
i j.
P (n)*(L) = E(u) T Pn(x) (uxN ), (6.21)
because the Atiyah-Hirzebruch spectral sequence can support no differential. He*
*re, x
is induced from the Chern class of the Hopf line bundle on CP N, which is a quo*
*tient
space of L, and u is uniquely defined as restricting to the standard generator *
*u1 2
P (n)*(S1), where we recognize the 1-skeleton L1 of L as the circle S1.
Since x is a Chern class, the coaction aeL is given on x by naturality as a*
*eLx = b(x).
Although L is not an H-space, there are, as in [ibid., (14.31)], partial multip*
*lications
L2kx L2m ! L on the skeletons whenever k + m = N, which imply that
n-1X i
aeLu = u 1 + xp a(i) in P (n)*(L) (6.22)
i=0
for certain elements a(i)2 2pi-1that this equation defines. (We warn that the*
*se
generators differ from W"urgler's [W"u77 ] and Yagita's [Ya77 ] generators ai b*
*y the
conjugation in ; as a result, certain formulae become transposed. Our generat*
*ors
are chosen for compatibility with [Bo95 ] and [Wi84 ], because they destabilize*
* properly
in xx7, 10.) The element a(n)does not exist because u fails to lift to the 2pn-*
*skeleton
of the lens space. As in [Bo95 , Thm. 14.32], the coalgebra structure is given *
*by
k-1X i
_Sa(k)= a(k) 1 + bp(k-i) a(i)+ 1 a(k) (6.23)
i=0
and fflSa(k)= 0.
Cohomology of real projective space, for p = 2 Here, the same test space
L is better known as real projective space R P 2N. It remains true that the Ati*
*yah-
Hirzebruch spectral sequence can support no differential, so that
P (n)*(R P 2N) = P (n)*[t]=(t2N+1), (6.24)
generated by the unique nonzero element t 2 P (n)1(R P 2N). As above, we find t*
*hat
n-1X i+1
aet = t 1 + t2 a(i), (6.25)
i=0
which defines elements a(i)2 2i+1-1. Indeed, this formula is identical to equ*
*a-
tion (6.22), since x = t2 is the Chern class of the complexified real Hopf line*
* bundle.
Thus equation (6.23) remains valid for p = 2.
Summary W"urgler [W"u77 ] and Yagita [Ya77 ] both proved that we now have
enough elements of to handle all stable operations.
- 23 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Theorem 6.26 The stable operations in P (n)-cohomology_are dual to the Hopf
algebroid = P (n)*(P (n), o) [replaced byP (n)*(P (n), o) if p = 2], which is*
* generated
as a P (n)*-algebra by the elements b(j)and a(i)defined by equations (6.14) and*
* (6.22)
[replaced by (6.25) if p = 2].
(a) For odd p, as a P (n)*-algebra,
= P (n)*(P (n), o) = E(a(0), a(1), . .,.a(n-1)) P (b(1), b(2), b(3)*
*, . .).;
(b) For p = 2, as a P (n)*-algebra,
____
=P (n)*(P (n), o) = P (a(0), a(1), . .,.a(n-1), b(n+1), b(n+2), . *
*.).,
and the elements b(j)for j n are given by the relations
a2(i)= b(i+1) for 0 i n - 1; (6.27)
(c) As a left P (n)*-module, is free with a basis consisting of all monomi*
*als
aIbJ = ai0(0)ai1(1).a.i.n-1(n-1)bj1(1)bj2(2)bj3(3).,. .
with multi-indices I = (i0, i1, . .,.in-1) and J = (j1, j2, . .).in which each *
*ir = 0 or 1;
(d) The right P (n)*-action on is given by multiplication by the elements *
*wk =
jRvk, where wn = vn1 and wk is determined inductively for k > n by the main rel*
*ation
(Rk) (see Definition 6.20);
(e) The comultiplication _S: ! is the P (n)*-algebra homomorphism given
on the generators by equations (6.15) and (6.23);
(f) The counit fflS: ! P (n)* is the P (n)*-algebra homomorphism given on *
*the
generators by fflSa(i)= 0 for all i and fflSb(j)= 0 for j > 0.
Proof What survives intact from W"urgler [W"u77 , Thm. 2.13] and Yagita [Ya77*
* ,
Lemma 3.5], even for p = 2, is (c) (using the conjugate generators to the a(i))*
*. Parts
(a), (d), (e) and (f) need no further comment.
[In (b), commutativity is not trivial; see Nassau [Na02 ] or [Bo ]. Since t*
*2 is a Chern
i+2
class, (aet)2 = ae(t2) = b(t2). Comparing the coefficients of t2 with the hel*
*p of (6.14)
and (6.25), we deduce (6.27) for i < n-1.
n+1
This argument fails for i = n-1, as t2 = 0; nevertheless, the result stil*
*l holds
by [Na02 , Thm. 2], which corrects [KW87 ]. Alternatively, the map of ring sp*
*ectra
P (n) ! P (n+1) in Lemma 2.7 sends each generator of (n) = to its namesake in
(n+1). As a2(n-1)= b(n)in (n+1), the only candidates for a2(n-1)in (n) are b*
*(n)
*
* _
and b(n)+ vn1. Since fflS(a2(n-1)) = (fflSa(n-1))2 = 0, we must choose b(n).] *
*|_|
7 Additive operations in P (n)-cohomology
In this section, we describe the additive unstable operations in P (n)-coho*
*mology
in the style of [BJW95 ], in terms of a certain bigraded algebra Q**, which, l*
*ike , is
a P (n)*-bimodule equipped with a coalgebra structure (_A, fflA) (called (Q(_),*
* Q(ffl))
in [BJW95 ]) that encodes the composition of operations and the identity opera*
*tion.
Although the results bear a strong formal resemblance to the stable results in *
*x6, the
stable proofs do not carry over; instead, one has to compute the whole Hopf rin*
*g in
x11 and then take the indecomposables.
- 24 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
i j
For p odd, we define Q**= QP (n)* P_(n)_*, the algebra of indecomposables in
i j
the Hopf ring P (n)* P_(n)_*. Specifically, Qkidenotes the group of indecomposa*
*bles
i j
in degree i of the Hopf algebra P (n)* P_(n)_k; its elements have total degree *
*i - k
in Q**(and this is the degree that governs signs). The multiplication and unit*
* in
Q**are induced from O-multiplication and the element [1] in the Hopf ring by the
homomorphisms (10.1). The left P (n)*-module action is induced from the Hopf ri*
*ng:
if v 2 P (n)j and c 2 Qki, we have vc 2 Qkj+i. [When p = 2, it should be noisur*
*prisej
after Proposition 6.12 that the correct Hopf ring to consider is not P (n)* P_(*
*n)_*
____ i j ____ i j
butP (n)* P_(n)_*; in this case, we set Q**= QP (n)* P_(n)_*. This is the same *
*left
i j
P (n)*-module as QP (n)* P_(n)_*, but with slightly different multiplication.]
By [RW96 , Cor. 1.5], both Q**and the Hopf ring are free P (n)*-modules. T*
*hese
conditions ensure [BJW95 , Lemma 4.16(a)] that the dual module to Q**is indeed*
* the
module of all additive unstable operations on P (n)-cohomology, and make availa*
*ble
all the machinery and results on additive operations. We thus identify:
(i)The additive unstable operation r: P (n)k(-) ! P (n)m (-);
` '
(ii)The primitive cohomology class r'k 2 P (n)m P_(n)_k;
(iii)The representing H-map of H-spaces r: P_(n)_k! P_(n)_m, up to homotop*
*y;
(iv)The P (n)*-linear functional : Qk*! P (n)*, of degree k - m.
The action of additive operations on P (n)*(X) is encoded in coactions
aeX : P (n)k(X) --! P (n)*(X) b Qk* (7.1)
(one for each k), which are monoidal as k varies [even if p = 2].
To construct the generators of Q**, we use the same test spaces as stably i*
*n x6,
together with the circle. We record the values of _A and fflA on each generator.
Cohomology of a point The right unit ring homomorphism jR: P (n)* ! Q*0is
just the coaction ae for the one-point space, and so is determined by the eleme*
*nts
k-1)
wk = jRvk 2 Q-2(p0 for k n. (7.2)
We use jR to make Q**a right P (n)*-module and the coactions aeX in (7.1) into*
* a
P (n)*-module homomorphism.
Cohomology of a circle The coaction for the circle S1 defines the suspension
element e 2 Q11by
aeu1 = u1 e in P (n)*(S1) Q1*= E(u1) Q1*. (7.3)
As in [BJW95 , Prop. 12.3(d)], _Ae = e e and fflAe = 1.
Then for any j > 0, the coaction for the j-sphere Sj is given by
aeuj = uj ej in P (n)*(Sj) Qj*= E(uj) Qj*. (7.4)
Given any additive operation r: P (n)k(-) ! P (n)m (-), representedibyjthe *
*map
r: P_(n)_k! P_(n)_m, where k, m > 0, we use P (n)k(Sj) ~=ssj P_(n)_k~= kP (n)j*
*-k to
- 25 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
rewrite the induced homomorphism on homotopy groups as
i j ss (r) i j
r*: kP (n)* ~=ss* P_(n)_k- --*-!ss* P_(n)_m ~= m P (n)*. (7.5)
By [BJW95 , Cor. 12.4], this is given on kv, where v 2 P (n)i, by the formula
r*( kv) = m . (7.6)
Complex orientation The coaction for CP 1 defines elements bj 2 Q22jby
1X
aex = b(x) = xj bj in P (n)*(C P 1) b Q2*~=Q2*[[x]], (7.7)
j=1
which is formally identical to equation (6.14), except that now b1 = e2 by [BJW*
*95 ,
Prop. 14.4(a)]. As in [ibid.], _Abi is the coefficient of xi in
1X
_Ab(x) = b(x)j bj in (Q** Q2*)[[x]], (7.8)
j=1
and fflAbj = 0 for j > 1.
Again [ibid., Lemma 14.6], we need only the accelerated elements b(j)= bpj *
*for
j 0, so b(0)= e2. The additive version of equation (6.19) also looks the same,
X1
b([p](x)) = [p]R(b(x)) = b(x)i+1jRgi in Q2*[[x]]. (7.9)
i=N
Definition 7.10 For each k n, we define the k-th main additive relation (Rk*
*) as
k
the coefficient of xp in equation (7.9).
In view of equation (6.17), the first two main relations are simply
n 2
(Rn) bp(0)wn = vnb(0) in Q* (7.11)
and n n+1
(Rn+1) bp(1)wn + bp(0)wn+1 = vn+1b(0)+ vpnb(1) in Q2*. (7.12)
We shall find in equation (10.10) that (Rn) desuspends once to
(R0n) ebN(0)wn = vne in Q1*. (7.13)
By equation (6.18), the general main relation for k n has the form
Xk i
(Rk) bp(k-i)wi 0 in Q2*mod V + W 2, (7.14)
i=n
where V = (vn, vn+1, vn+2, . .).and W = (wn, wn+1, wn+2, . .).denote ideals in*
* Q**.
Cohomology of a lens space Our last test space is the lens space skeleton L,
whose cohomology is given by equation (6.21), assuming p is odd. We already know
aeLx = b(x) from equation (7.7). For u, we find, as in [BJW95 , (16.21)], that
n-1X i
aeLu = u e + xp a(i) in P (n)*(L) Q1* (7.15)
i=0
- 26 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
for certain elements a(i)2 Q12pithat this equation defines. We deduce that
Xk i
_Aa(k)= a(k) e + bp(k-i) a(i) (7.16)
i=0
and fflAa(k)= 0.
[If p = 2, L = R P 2N has different cohomology (6.24), and we replace equa-
tion (7.15) by
n-1X i+1
aet = t e + t2 a(i) in P (t)=(t2N+1) Q1*. (7.17)
i=0
Nevertheless, equation (7.16) and fflAa(k)= 0 remain valid for p = 2. By [Bo ],*
* equa-
tion (6.27) destabilizes in the obvious way, to
a2(i)= b(i+1) for 0 i n - 1.] (7.18)
More relations We shall find in equation (10.14) that one more suspension fact*
*or
can be squeezed out of (7.13) if we first multiply by a(0), to give
(R00n) a(0)bN(0)wn = vna(0) in Q1*. (7.19)
[When p = 2, we can multiply this by another a(0)and use equation (7.18) to obt*
*ain
the unexpected formula
bN(0)b(1)wn = vnb(1). (7.20)
This is not all; if we multiply (Rn+1) (7.12) by bN(0), we obtain the reduction*
* formula
n+1 2 N 2n 2n
bN+2(0)wn+1 = vnb(0)b(1)+ vnb(1)+ vn+1b(0), (7.21)
by using (7.20) to simplify one of the terms.]
Summary We have the additive version of the Hopf algebroid .
Theorem 7.22 The additive unstableioperationsjin P (n)-cohomology_areidualj*
*to
the P (n)*-algebra Q**= QP (n)* P_(n)_* [replaced by QP (n)* P_(n)_*if p = 2],
which has the properties:
(a) Q**is the commutative bigraded P (n)*-algebra generated by the elements:
k-1)
wk 2 Q-2(p0 for k n, defined by jR in equation (7.2);
e 2 Q11, the suspension element, defined by equation (7.3);
b(j)2 Q22pjfor j 0, defined by equation (7.7);
a(i)2 Q12pifor 0 i < n, defined by (7.15) [replaced by (7.17) if p =*
* 2];
subject to the relations e2 = b(0), the main relations (Rk) for k > n (see Defi*
*ni-
tion 7.10), and the two variants (7.13) and (7.19) of (Rn) [also (7.18) if p = *
*2];
(b) Q**is a free left P (n)*-module;
(c) Multiplication by the elements wk makes Q**a right P (n)*-module;
(d) The comultiplication _A: Q**! Q** Q**is the homomorphism of algebras and
of P (n)*-bimodules given on each generator as noted above;
(e) The counit fflA: Q**! P (n)* is the P (n)*-algebra homomorphism given on
generators by fflAe = 1, fflAa(i)= 0, fflAb(j)= 0 for j > 0, fflAb(0)= 1, and f*
*flAwk = vk.
- 27 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Parts (c), (d) and (e) need no further comment. Part (b) is included in Th*
*eo-
rem 8.4. Part (a) can be read off from Theorem 11.1. [For commutativity when p *
*= 2,
we refer to [Bo ].]
We recall [BJW95 , (6.3)] the stabilization homomorphism Q(oe): Qk*! , wh*
*ich
has degree zero. We may use it to recover the structure on in Theorem 6.26 fr*
*om
Q**simply by setting e = 1. The coalgebra structure (_A, fflA) stabilizes to (_*
*S, fflS).
8 Relations for additive operations
We noted in Theorem 7.22 that Q**is a free P (n)*-module, which is not at a*
*ll
obvious from the generators and relations given. In this section, we exhibit a *
*basis of
Q**and prove in Lemma 8.5 that it spans the module.
We also establish some direct applications of additive operations.
The Ravenel-Wilson basis Since e2 = b(0)and a2(i)= 0 trivially if p is odd
[replaced by a2(i)= b(i+1)if p = 2, from equation (7.18)], any monomial in the *
*listed
generators of the P (n)*-algebra Q**can be written in the abbreviated form
efflaIbJwK = efflai0(0)ai1(1).a.i.n-1(n-1)bj0(0)bj1(1)bj2(2).w.k.nnw*
*kn+1n+1wkn+2n+2.,.(.8.1)
with multi-indices I = (i0, i1, . .,.in-1), J = (j0, j1, j2, . .)., and K = (kn*
*, kn+1, . .).,
where each ir, also ffl, is 0 or 1. (We keep the w's to the right, as a reminde*
*r that they
define the right action of P (n)* on Q**.) We introduce the following parameter*
*s:
P
The b-length is rjr, the total number of factors of the form b(j);
P
The w-length is rkr, the total number of factors of the form wk.
As with BP in [RW77 ], it is easier to specify which monomials are not wan*
*ted in
forming the basis than those which are. [For p = 2, the basis is not written ou*
*t in
detail in [RW96 ], and contains some surprises.] There are two variants; we sh*
*all need
the second in xx10, 11.
Definition 8.2 We call the monomial (8.1) Q-allowable if it does not have any*
* of
the following forms [note that (iv) and (v) apply only if p = 2]:
n pn+1 pq
(i)bp(dn)b(dn+1). .b.(dq)wqc, with 0 dn dn+1 . . .dq, q n;
n+1 pq
(ii)ebN(0)bp(dn+1).b.(.dq)wqc, with 0 dn+1 . . .dq, q n;
n+1 pq
(iii)a(0)bN(0)bp(dn+1).b.(.dq)wqc, with 0 dn+1 . . .dq, q n;
n+1 2q (8.3)
(iv)bN(0)b(1)b2(dn+1).b.(.dq)wqc, where p = 2, with 0 dn+1 . . .dq,
q n;
n+1 2n+2 2q
(v) bN+2(0)b(dn+2). .b.(dq)wqc, where p = 2, with 0 dn+2 . . .dq,
q n + 1;
where c is any monomial (c = 1 is permitted) in the generators e, a(i), b(j), a*
*nd wk.
More generally, we call the monomial allowable if it is not of the form (i)*
* or (ii).
- 28 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Remark In [RW96 ], a monomial is called n-allowable (lies in An) if it is no*
*t of the
form (i). If it contains a factor e or a(0), it is called n-plus allowable (lie*
*s in A+n) if it
is not of the form (i), (ii) or (iii).
From [RW96 , Thm. 1.3], we have the Ravenel-Wilson basis of Q**.
Theorem 8.4 (Ravenel-Wilson) TheiQ-allowablejmonomials_(8.1)iformja basis of
the free P (n)*-module Q**= QP (n)* P_(n)_*[or QP (n)* P_(n)_*if p = 2].
Later in this section, we shall reprove half the theorem.
Lemma 8.5 The relations e2 = b(0), the main relations (Rk) for k > n, [relat*
*ion
(7.18) if p = 2,] and the variants (7.13) and (7.19)iof (Rn)jimply_that_theiQ-a*
*llowablej
monomials (8.1) span the P (n)*-module Q**= QP (n)* P_(n)_* [or QP (n)* P_(n)_*
if p = 2].
Generators of cohomology Just as in [BJW95 , Thm. 20.2], Theorem 1.16 follows
directly from the fact that the additive operations on P (n)-k(-) form the P (n*
*)*-dual
of the free P (n)*-module Q-k*, whose generators all lie in groups Q-kjwith j *
* 0.
We combine the following two lemmas, which correspond to Theorem 20.3 and
Lemma 20.5 of [ibid.]. We study the linear functional fflA = <'-k, ->: Q-k*! P*
* (n)*
defined by the identity operation '-k on P (n)-k(-), which is plainly additive.
Lemma 8.6 Given any integer k > 0, there exist:
(i)a sequence of additive unstable operations ri: P (n)-k(-) ! P (n)m(i)(*
*-)
with m(i) 0;
(ii)a sequence of elements v(i) 2 P (n)* with deg(v(i)) ! 1;
such that in any additively unstable P (n)-cohomology comodule M (e. g. P (n)*(*
*X)
for any space X), any x 2 M-k decomposes as the (topological infinite) sum
X
x = v(i)rix.
i
Proof Let {c1, c2, c3, . .}.be the Ravenel-Wilson (or any other) basis of the*
* free
P (n)*-module Q-k*, with ci2 Q-km(i). Trivially, m(i) 0. For fixed x 2 M-k an*
*d any
additive operation r, the linearity of rx in r may be expressed, as in [BJW95 *
*, (6.39)],
by the formula X
rx = rix, (8.7)
i
where ri denotes the operation dual to ci. We take r = '-k, put v(i) = <'-k, ci*
*>, and
_
note that deg(v(i)) = m(i) + k ! 1. |_|
Remark The coefficients are readily computed: v(i) = fflAci= vK if the monomi*
*al ci
has the form efflbj(0)wK , and v(i) = 0 otherwise. Thus many terms are zero.
To get the more precise information for Theorem 1.16, we write the space X *
*as
the disjoint union of its components and reduce to the case when X is connected.
- 29 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Lemma 8.8 Let M be a connected (see [BJW95 , Defn. 7.14]) additively unstab*
*le
P (n)-cohomology algebra (e. g. P (n)*(X) for any connected space X). Then as a
topological P (n)*-module, M is generated by 1M 2 M0 and elements of Mi for i *
*> 0.
The generator 1M is never redundant.
Proof Let L be the submodule generated (topologically) by the elements of all*
* the
Mi for i > 0. By Lemma 8.6, we need only consider x 2 M0. We choose a basis
{c1, c2, c3, . .}.of Q0*with c1 = 1.
We recall from [BJW95 , Defn. 7.13] the collapse operation ~j on P (n)j(-)*
* for any
j; since M is connected, on any x 2 Mj it satisfies ~jx = v1M for some v 2 P (*
*n)-j.
But (8.7) gives ~0x r1x mod L and also x = '0x r1x mod L. Thus x ~0x =
~1M mod L for some ~ 2 Fp.
_
Since ~L = 0 and ~01M = 1M , 1M never lies in L. |_|
Higher-order relations The proof of Lemma 8.5 resembles that of [BJW95 ,
Thm. 18.16]. The Nakayama Lemma [Bo95 , x15] (which is easier for P (n)* than
for BP*, as p = 0) allows us to work throughout modulo the ideal V Q**. We al*
*so
work modulo powers of W . (These ideals were introduced in equation (7.14), whi*
*ch
displays the w-linear terms in the relation (Rk).)
When q = n and c = 1, we observe that (8.3)(i) is the first term in (Rdn+n)*
*, and is
thus expressible by equation (7.14) in terms of Q-allowable monomials mod V + W*
* 2.
Equation (7.13) shows that (R0n) takes care of (ii), while (7.19) shows that (R*
*00n) takes
care of (iii). [If p = 2, we use (7.20) and (7.21) to handle (iv) and (v).]
Otherwise, the relations (Rk) are not at all transparent. We handle the gen*
*eral
disallowed monomial (8.3)(i) by eliminating the q - n variables wn, wn+1, . .,.*
*wq-1
from the q -n+1 relations (Rdn+n), (Rdn+1+n+1), . . . , (Rdq+q), expressed in t*
*he form
(7.14), to obtain the higher-order derived relation
X 2
qwq + rwr 0 mod V + W , (8.9)
r>q
for certain determinants r. Explicitly, for any r q,
X pssn pss(q-1) pssr
r = fflssb(dn+n-ssn).b.(.dq-1+q-1-ss(q-1))b(dq+q-ssr),(8.10)
ss
where we sum over all permutations ss of {n, . .,.q-1, r}, write fflssfor the s*
*ign of ss,
and adopt the convention that meaningless factors b(j)with j < 0 are taken as 0.
We order the b -monomials lexicographically (bJ < bK if and only if there *
*exists
t 0 such that jr = kr for all r < t, and jt< kt).
Lemma 8.11 For any r q, the determinant r in (8.10) has the form
n pn+1 pq-1 pr
r = bp(dn)b(dn+1). .b.(dq-1)b(dq+q-r) + higher terms.
Proof The displayed term is the diagonal term with ss = id. For any other per*
*mu-
tation ss, there is a first index t such that sst > t, so that n t q - 1 an*
*d ssk = k
n pt-1 psst
for all k < t. The corresponding term fflssbp(dn).b.(.dt-1)b(dt+t-sst).i.n.(8.1*
*0) is higher,
_
because dt+ t - sst < dt. |_|
- 30 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Proof of Lemma 8.5 We show that each Q-disallowed monomial in (8.3) is a linear
combination mod V of higher monomials with the same w-length, and monomials of
greater w-length, where we partially order all monomials according to the facto*
*r bJ
(and ignore e, a(i), and wk). Since there are only finitely many monomials in *
*each
bidegree, the result follows. n q
For (8.3)(i), Lemma 8.11 shows that we can use (8.9) to express bp(dn).b.p.*
*(dq)wq
as a linear combination mod V of higher monomials and monomials with w-length
2, since for r > q, the diagonal term of r is higher than the diagonal term *
*of q.
Multiplication by c preserves the ordering.
For (ii), (iii) [and (iv), if p = 2], we modify equation (8.9) by eliminati*
*ng the
variables wn+1, . . . , wq-1 from the relations (Rdn+1+n+1), . . . , (Rdq+q) to*
* obtain
X 2
0nwn + 0qwq + 0rwr 0 mod V + W .
r>q
When we multiply by ebN(0)c, the first term drops out by (7.13). Lemma 8.11, sl*
*ightly
modified (or with n replaced by n + 1), shows that (ii) is the lowest of the re*
*maining
terms. If we multiply by a(0)bN(0)c instead and use (7.19), we obtain (iii). [F*
*or (iv), we
multiply by bN(0)b(1)c and use (7.20).]
[For (v), we eliminate the variables wn+2, . .,.wq-1 from the relations (Rd*
*n+2+n+2),
. . . , (Rdq+q) to obtain a higher-order relation
X 2
00nwn + 00n+1wn+1 + 00qwq + 00rwr 0 mod V + W .
r>q
n+1
When we multiply this by bN+2(0)c, the first two terms drop out by (7.11) and (*
*7.21).
_
The diagonal term in the determinant 00qgives (v).] |_|
The first higher-order relation The first relation for a given q, where we eli*
*mi-
nate wn, wn+1, . . . , wq-1from (R0n), (Rn+1), . . . , (Rq), is particularly im*
*portant. The
additive version for P (n) of Bendersky's Lemma [Be86 , Thm. 6.2] (or see [BJW9*
*5 ,
Lemma 18.23]) gives more precise information than our proof of Lemma 8.5, and
follows immediately from Lemma 12.2.
We recall the ideal Iq = (vn, vn+1, . .,.vq-1) P (n)* (where In = (0)).
i j ____ i j
Lemma 8.12 In Q**= QP (n)* P_(n)_* [replaced by QP (n)* P_(n)_*if p = 2], we
have the relation
eg(n,q)-1wq vqeg(n,q-1)+1modIqQ** for q n. (8.13)
[If p = 2, this is almost superseded by the relation
_
eg(n,q)-2wq vqeg(n,q-1)modIqQ** for q n + 1.]|_| (8.14)
Primitive elements Let M be an unstable P (n)-cohomology comodule (in the
sense of [BJW95 , Defn. 6.32]). An element x 2 Mk is called (additively unsta*
*bly)
primitive if the coaction aeM has the value aeM x = x ek on x. Then for any v 2*
* P (n)*,
aeM (vx) = x ek(jRv). (8.15)
Of course, all this requires k 0, but more is true, as in [ibid., Lemma 2*
*0.8].
- 31 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Lemma 8.16 Let x 2 Mk be a nonzero primitive element of the unstable P (n)-
cohomology comodule M, and assume q n.
(a) If Iqx = 0 and k satisfies the condition (depending on p and q):
(i)k g(n, q) - 1 if p is odd or q = n;
(8.17)
(ii)k g(n, q) - 2 if p = 2 and q n + 1;
then vqx is primitive (possibly zero);
(b) If k does not satisfy the condition (8.17), then for all i > 0, viqx is *
*nonzero and
is not primitive.
Proof For (a), (8.15) gives ae(vqx) = x ekwq. By Lemma 8.12, this is the sa*
*me as
q-1) k-2(pq-1)
x vqek-2(p = vqx e , since Iqx = 0.
For (b), we have ae(viqx) = x ekwiq. Here, ekwiqis Q-allowable by (8.3) a*
*nd hence
_
a basis element of Q**, which shows that viqx is not primitive. |_|
Proof of Lemma 1.17 We must have aex = x ek. If m > n, we have vm-1 x = 0, a*
*nd
case (b) of Lemma 8.16 with q = m - 1 does not apply; hence the lower bound on *
*k.
Conversely, (8.15) specifies the coaction on all of M, and Lemma 8.12 shows*
* it is
_
well defined. |_|
Proof of Theorem 1.18 We build an increasing sequence
0 = M0 M1 M2 . . .M
of subcomodules of M. For each i > 0, just as in the proof of Theorem 20.11 in
[BJW95 ], we construct a primitive element xi2 M=Mi-1with Ann (xi) = Imi for s*
*ome
mi, using Lemma 8.16 in place of Lemma 20.8 of [ibid.]. We take Mi=Mi-1 M=Mi-1
as the P (n)*-submodule generated by xi. Lemma 1.17 describes Mi=Mi-1.
Because each ki 0 in Theorem 1.18 and each Mk is a finitely generated Fp-
module, this sequence must terminate after finitely many steps. We deduce that M
_
is a finitely presented P (n)*-module. |_|
9 Idempotent operations
Lemma 9.1 delivers the promised additive idempotent operations `(m) in P (n*
*)-
cohomology that we need for Lemma 5.1, which is equivalent to Lemma 3.1. In fac*
*t,
we find a large class of `(m), among which none seems to be preferred. The rest*
* of
this section applies the work in x8 to prove Lemma 9.1.
Lemma 9.1 Assume that k g(n, m) [replaced by k g(n, m) - 1 if p = 2], wh*
*ere
m n. Then there exists an additive idempotent operation `(m) on P (n)k(-) hav*
*ing
the following properties:
(i)The image of the operation `(m) is represented by the space P_(n,_m)_k;
(ii)The_map `(m): P_(n)_k! P_(n)_kfactors to yield an H-space splitting
`(m) : P_(n,_m)_k! P_(n)_kof the canonical H-map ae(m): P_(n)_k! P_(n,_*
*m)_k;
____
(iii)For all spaces X, `(m) naturally embeds P (n, m)*(X) P (n)*(X) as a
summand, in the sense of abelian groups (but not as P (n)*-modules).
- 32 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Remark Exactly as in [BJW95 , end of x22], we can make the splittings `(m) c*
*om-
patible as k and m vary if we wish. The decomposition factors of P_(n)_kresult*
*ing
from this approach must of course be the same as in Theorem 1.12, according to
Theorem 1.2(b), but the injection maps are different, in general.
However, we emphasize that the splitting theorems as stated in xx1, 5 do not
require any compatibility.
The ideals Jm As in [BJW95 ], the ideal Jm = (vm+1 , vm+2 , . .). P (n)*, in*
*troduced
in equation (1.8), gives rise to an analogous ideal for the right action of P (*
*n)* on Q**.
Definition 9.2 Given any m n, we define the ideal
Jm = (wm+1 , wm+2 , wm+3 , . .). Q**.
We need to know how Jm sits inside Q**. As in [ibid.], the answer is remark*
*ably
clean, in a certain range.
Lemma 9.3 For k g(n, m) [replaced by k g(n, m) - 1 if p = 2], Qk*\ Jm is*
* the
left P (n)*-submodule of Qk*spanned by all the Q-allowable monomials (8.1) that*
* lie
in it and contain an explicit factor wq for some q > m.
Remark By Lemma 8.12, vm+1 ebg(n,m)=2(0)- z lies in Jm , where z 2 Im+1 Q**, *
*so the
result definitely fails for k = g(n, m) + 1 [also for k = g(n, m) if p = 2].
Proof Any monomial that contains wh with h > m visibly lies in Jm . To show t*
*he
converse, we fix k and i0 and prove by downward induction on h that for all i *
* i0, all
elements in Qkiof the form cwh lie in the indicated P (n)*-submodule. This stat*
*ement
is trivial for sufficiently large h (depending on k and i0).
We therefore choose q > m, assume the statement holds for allqh > q, and pr*
*ove
it for h = q. Take cwq 2 Qki, where i i0, so that c 2 Qk+2(pi-1). By Lemma 8.*
*5, we
may reduce to the case where c is a Q-allowable monomial. We note that in (8.3*
*),
the Q-disallowed monomials (i) and (iv) have b -length 1_2g(n, q), while (ii), *
*(iii) and
(v) have b -length 1_2g(n, q) - 1.
Case 1: c has no factor e, a(0), or wj. For odd p, the b -length of c is at*
* most
1_ q 1_ q 1_
2(k + 2(p -1)) 2(g(n, m) + 2p - 2) < 2g(n, q),
which makes cwq also Q-allowable, as only rule (i) of (8.3) is relevant. [If p *
*= 2, we
need to assume k g(n, m) - 1 to get the stronger bound 1_2g(n, q) - 1.]
Case 2: c = ey or c = a(0)y, where y has no factor wj. In this case, the b *
*-length
of c is at most
1_ q 1_ q 1_
2(k - 1 + 2(p -1)) 2(g(n, m) + 2p - 3) < 2g(n, q) - 1,
which makes cwq automatically Q-allowable.
Case 3: c = ywj, where j q. Then cwq remains Q-allowable, by the form of
Definition 8.2.
Case 4: c = ywj, where j > q. By induction, cwq = (ywq)wj lies in the indic*
*ated
_
submodule. |_|
Linear functionals To establish Lemma 9.1, we actually construct the associated
P (n)*-linear functional <`(m), ->: Qk*! P (n)*.
- 33 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Lemma 9.4 Assume the linear functional <`(m), ->: Qk*! P (n)* corresponding *
*to
the additive operation `(m): P (n)k(-) ! P (n)k(-) satisfies the conditions:
(i) <`(m), Qk*\ Jm> = 0;
(9.5)
(ii) <`(m), c> fflAc mod Jm for all c 2 Qk*;
where fflA: Qk*! P (n)* is the augmentation. Then:
(a) The homology homomorphism Q(`(m)*): Qk*! Qk*induced by the represent-
ing map `(m): P_(n)_k! P_(n)_ksatisfies
(i) Q(`(m)*)(Qk*\ Jm) = 0;
(ii) Q(`(m)*) id: Qk*! Qk*mod J m;
(b) Q(`(m)*) induces a splitting of the short exact sequence
0 --! Qk*\ Jm --! Qk*--! Qk*=(Qk*\ Jm) --! 0
of left P (n)*-modules;
(c) The operation `(m) is idempotent and has the properties listed in Lemma *
*9.1.
Proof The proof is patterned after that of Lemma 22.2 in [BJW95 ]. We requir*
*e the
commutative diagram
_A id <`(m),-> ~R
Qk*_____________-Q** Qk* _________-Q** P (n)*____________-Q**
| | * | |
|q0 | | |q0
| | | |
|? ___ |? __ |? __ |?
A id fflA ~R
Qk*=Jm __________Q**- Qk*=Jm ______Q**- P (n)*=Jm ________-Q**=Jm
__ __ __
of P (n)*-module homomorphisms, where_A , fflAand~R denote quotients of _A, f*
*flA,
and the right action ~R of P (n)* on Q**, Qk*=Jm is really Qk*=(Qk*\ Jm), and *
*the
vertical arrows are the obvious projections. The conditions (9.5) on <`(m), ->*
* are
exactly what we need to fill in the diagonal.
By [BJW95 , Lemma 6.51(c)], the top row gives the homology homomorphism
Q(`(m)*), while by [ibid., (6.31)], the bottom row reduces to the identity homo*
*mor-
phism of Qk*=Jm . Thus the diagonal provides a splitting we call j0: Qk*=Jm !*
* Qk*
that satisfies j0Oq0= Q(`(m)*) and q0Oj0 = idand so yields (a). Part (b) is mer*
*ely a
restatement of (a).
It follows by faithfulness that `(m) is an idempotent operation, so that th*
*e image
h(-) = `(m)P (n)k(-) P (n)k(-) is an ungraded cohomology theory. By [Bo95 ,
Thm. 3.6], h(-) is represented (on Ho ) by some H-space Y , and the additive op*
*er-
ations h(-) P (n)k(-) and `(m): P (n)k(-) ! h(-) are represented by H-maps
j: Y ! P_(n)_kand q: P_(n)_k! Y respectively, that satisfy jOq = `(m) and qOj =*
* id.
To finish (c),iwejapply the homotopy group functoriss*(-)jto obtain homomor-
phisms q*: ss* P_(n)_k ! ss*(Y ) and j*: ss*(Y ) ! ss* P_(n)_k that satisfy q*O*
*j* = id
i j
and j*Oq* = `(m)*. Recall that ss* P_(n)_k ~= kP (n)*. Given v 2 P (n)i, (7.*
*6)
- 34 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
evaluates `(m)* kv = k<`(m), ek+i(jRv)>. Then (9.5)(i) yields `(m)* kv = 0
if v 2 Jm , while for any v, (ii) gives `(m)* kv kv mod Jm . It follows that
ae(m)Oj: Y ! P_(n,_m)_kinduces an isomorphism of homotopy groups and is therefo*
*re
a_homotopy_equivalence. To establish the properties listed in Lemma 9.1, we put
_
`(m) = jOg, where g: P_(n,_m)_k! Y is a homotopy inverse to ae(m)Oj. |_|
Proof of Lemma 9.1 Lemma 9.3 makes it obvious that linear functionals <`(m), ->
_
exist that satisfy the conditions (9.5), so that Lemma 9.4 applies. |_|
Remark As an explicit example, choose <`(m), -> on the Ravenel-Wilson basis as
<`(m), c> = vK if c has the form efflbj(0)wK but contains no factor wk with k >*
* m, and
<`(m), c> = 0 otherwise. To determine <`(m), c> for c not in the basis, we must*
* first
express c in terms of the basis.
10 Unstable operations in P (n)-cohomology
In this section, we use all unstable operationsiin Pj(n)-cohomology to obt*
*ain
generators and relations for the Hopf ring P (n)* P_(n)_*, in the style of [BJW*
*95 ]. The
two multiplications are c*d = ~*(cxd) and cOd = OE*(cxd), induced respectively *
*by the
maps ~: P_(n)_kx P_(n)_k! P_(n)_kand OE: P_(n)_kx P_(n)_m! P_(n)_k+mthat repres*
*ent
addition and multiplicationiinjP (n)-cohomology, and 1k will denote_thei*-ident*
*ityj
element of P (n)* P_(n)_k. [If p = 2, we use the Hopf ring P (n)* P_(n)_*inste*
*ad,
__
replacing c x d by c x d in both multiplications.] We still assume that 0 < n <*
* 1.
We deduce the results of x7 on additive operations by applying the homomorp*
*hism
i j
qk: P (n)* P_(n)_k--! Qk*, (10.1)
which neglects 1k and decomposables, shifts degrees by -k, and (as k varies) ta*
*kes
O-products to products (with a sign, on account of the degree shift). However,*
* the
Hopf ring structure maps _ and ffl are unrelated to _A and fflA.
Since the Hopf ring is a free P (n)*-module by [RW96 , Cor. 1.5], Theorem *
*4.14 of
[BJW95 ] allows us to identify:
(i)The cohomology operation r: P (n)k(-) ! P (n)m (-);
i j
(ii)The cohomology class r('k) 2 P (n)m P_(n)_k;
(iii)The representing map of spaces r: P_(n)_k! P_(n)_m, up to homotopy;
i j
(iv)The P (n)*-linear functional : P (n)* P_(n)_k ! P (n)* of degree*
* -m
____ i j
[or :P (n)* P_(n)_k! P (n)* if p = 2].
i j ____ i j
Hopf rings for p = 2 When p = 2, P (n)* P_(n)_*andP (n)* P_(n)_*are not Hopf
i j
rings in the ordinary sense (though H* P_(n)_*; F2 is one, and is described in *
*[BW01 ]
and after Theorem 11.8). (A few things are simpler: there are no signs and O is*
* the
identity.) Because P (n) is not commutative, all Hopf ring axioms that shuffle *
*factors
must be modified to use the commutativity isomorphism TQ of equation (2.11), wh*
*ich
results in extra terms; see x2 or [Bo ] for details. Neither multiplication is *
*commutative
in the ordinary sense, nor is _ cocommutative.
- 35 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
____i j i *
* j
As a P (n)*-module, the Hopf ring P (n)*P_(n)_* is identical to P (n)* P_(n*
*)_*.
The_choice_ofimultiplicationjon P (n) does not affect the P (n)*-module structu*
*re on
P (n)* P_(n)_*, nor does it affect the O-generators that we construct below. Ho*
*wever,
__
switching to the other good multiplication on P (n) replaces cOd = OE*(cx d) by
_ __
OE*(cxd) = OE*T*(cxd) = OE*(dx c) = dOc,
which is different in general; and similarly for c * d.
The Cartan formulae Assume first that p is odd. Given a cohomology class
x 2 P (n)k(X), we encode the action of operations on x by a formula of the form
X
r(x) = xff for all r,
ff
i j
for suitable choices cff2 P (n)* P_(n)_kand xff2 P (n)*(X). (Here and elsewhere*
*, we
mean all operations r that have the correct domain degree. The sum may be infin*
*ite
if X is not finite-dimensional.) Similarly, given y 2 P (n)m (X), suppose
X
r(y) = yfi for all r.
fi
Then the two Cartan formulae [BJW95 , (10.23) and (10.36)] are:
X X
r(x + y) = (-1)deg(xff) deg(yfi)xffyfi(10.2)
ff fi
and X X
r(xy) = (-1)deg(xff) deg(yfi)xffyfi. (10.3)
ff fi
We use them repeatedly without further reference.
The case p = 2 Examination reveals that the proof of the Cartan formulae in
[BJW95 ] relies on the identity_ = , which is false for*
* p = 2;
we must replace_aix b byja x b and use equation (2.15) instead. When we use the
Hopf ringP (n)* P_(n)_*, both Cartan formulae remain valid as stated.
Cohomology of a point Our first test space isithe one-pointjspace._iFor eachj
v 2 P (n)q, the Hopf ring element [v] 2 P (n)0 P_(n)_-q [orP (n)0 P_(n)_-q if p*
* = 2]
is defined by the identity
r(v) = in P (n)*(point) = P (n)*, for all r. (10.4)
The properties of these elements were listed in [BJW95 , Prop. 11.2]. As [v +*
* v0] =
[v] * [v0] and [vv0] = [v]O[v0], we are primarily interested in the elements
i j
[vk] 2 P (n)0 P_(n)_-2(pk-1) for k n
____ i j
[or inP (n)0 P_(n)_-2(2k-1)if p = 2]. Then (10.1) maps [vk] to wk.
We have the important relation
[1]*p= [p] = [00] = 10. (10.5)
- 36 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Cohomology of a circle iOur secondjtest_spaceiisjthe circle S1. The suspension
element e = e1 2 P (n)1 P_(n)_1 [orP (n)1 P_(n)_1 if p = 2] is defined by the a*
*ction
of operations r on the standard generator u1 2 P (n)1(S1),
r(u1) = 1S + u1 in P (n)*(S1) = E(u1), for all r.(10.6)
The properties of e were listed in [BJW95 , Prop. 13.7].
Complexiorientationj_Our_thirditestjspace is CP 1. The Hopf ring elements bj 2
P (n)2j P_(n)_2[orP (n)2j P_(n)_2if p = 2] for j 0 are defined by the identity
1X
r(x) = = xj in P (n)*(C P 1) ~=P (n)*[[x]], for(all1r*
*,0.7)
j=0
P j
where b(x) is a convenient formal abbreviation for jbjx . Their properties w*
*ere
listed in [BJW95 , Prop. 15.3]. In particular, b0 =i12 isjnow nonzero_andib1 =*
* -eOe.j
Again, the accelerated elements b(j)= bpj2 P (n)2pj P_(n)_2[or inP (n)2j+1 P_(n*
*)_2
if p = 2] suffice, as [ibid., Lemma 15.9] shows how to express the other b's in*
*ductively
in terms of these and the v's and [v]'s.
Naturality of equation (10.7) with respect to the p-th power map i: CP 1 ! *
*CP 1,
with massive use of the Cartan formulae, yields the identity
1 i j
b([p](x)) = * {b(x)Oi+1O[gi]} in P (n)* P (n) [[x]] (10.8)
i=N _____2
____ i j
[or inP (n) *P_(n)_2[[x]] if p = 2], as in [ibid., (15.14)]. The lowest power o*
*f x that
n
occurs is still xp , apart from the term 12 on each side.
Definition 10.9 For each k n, we define the k-th main unstable relation (Rk*
*) as
k
the coefficient of xp in equation (10.8).
The first relation is simply
n i j
(Rn) vnb(0)= bOp(0)O[vn] in P (n)* P_(n)_2
____ i j
[or inP (n)* P_(n)_2 if p = 2]. By [RW96 , Prop. 2.1(j)], it desuspends once to
i j
(R0n) vne = eObON(0)O[vn] in P (n)* P_(n)_1 (10.10)
____ i j
[or in P (n)* P_(n)_1if p = 2]. The second relation is almost as easy, in view*
* of
equation (6.17):
n Opn+1 p
(Rn+1) bOp(1)O[vn] + b(0) O[vn+1] = vnb(1)+ vn+1b(0).(10.11)
Cohomology of a lens space, for p odd Our final test space is the lens space
skeleton L, whose cohomology (6.21) has two generators u and x. As x is a Chern
class, equation (10.7) gives r(x) by naturality. We define Hopf ring elements a*
*i and
ci by the identity
NX N-1X
r(u) = xi+ uxi in P (n)*(L), for all r. (10.12)
i=0 i=0
- 37 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
Not by coincidence, their formal properties are exactlyPthe same as in the case*
* E =
K(n) of [BJW95 ]. The formal abbreviation a(x) = iaixi is convenient.
i j
Proposition 10.13 For p odd, the Hopf ring elements ai 2 P (n)2i P_(n)_1(for
i j i *
* j
0 i < pn), a(i)= api2 P (n)2piP_(n)_1 (for 0 i < n), and ci 2 P (n)2i+1 P_(*
*n)_1
(for 0 i pn - 2) defined by equation (10.12) have the following properties:
(a) a0 = 11 and c0 = e;
P
(b) _ak = i+j=kai aj;
(c) fflai= 0 for all i > 0, in particular, ffla(i)= 0 for all i;
i j
(d) ai* aj = i+jiai+j, provided i + j < pn;
(e) a*p(i)= 0 for 0 i < n - 1;
(f) Oai= (-1)iai, in particular, Oa(i)= -a(i);
(g) ci= e * ai;
(h) a(i)Oa(j)= -a(j)Oa(i);
(i) a(i)Oa(i)= 0;
(j) For all r, r*ak is the coefficient of xk in the formal identity
N N-1
r*a(x) = * {b(x)OiO[]} * * {a(x)Ob(x)OiO[]}
i=0 i=0
i j n
in P (n)* P_(n)_*[x]=(xp ).
Proof The statement and proof are identical to [BJW95 , Prop. 17.16], except*
* that
we offer a simpler proof of (f) (and could have also in [ibid.]; compare the di*
*vided
power Hopf algebra (a1)).
If m is odd, say m = 2k + 1, we can write the defining equation for Oam as
Xk
Oam + (Oam-i * ai+ Oai* am-i) + am = 0.
i=1
By induction, the terms in the sum cancel in pairs, as m-i and i have opposite *
*parity.
*
* _
If m is even, (d) decomposes am as a *-product, and we again use induction.*
* |_|
We emphasize that (e) is not valid for i = n - 1; instead, [RW96 , Prop. 2*
*.1(i)]
shows that the unstable analogue of equation (7.19) is
i j
(R00n) a*p(n-1)= vna(0)- a(0)ObON(0)O[vn] in P (n)* P_(n)_1.(10.1*
*4)
Cohomology of real projective space, for p = 2 In this case, L = RP 2N, with
cohomology (6.24). We define Hopf ring elements fi by the identity
2NX
r(t) = ti in P (n)*(R P 2N) = P (n)*[t]=(t2N+1), for all(r.10.*
*15)
i=0
Again, we mimic equation (10.12) by writing ai = f2i, a(i)= a2i= f2i+1, andPci =
f2i+1. We make the obvious changes to Proposition 10.13 and write f(t) = ifi*
*ti.
- 38 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
We warn that the analogy is not perfect; _ak acquires many extra terms. Also, (*
*d)
now requires proof; see [Bo ].
____i j
Proposition 10.16 For p = 2, the Hopf ring elements fi 2 P (n)iP_(n)_1 (for
____ i j
0 i 2N) and a(i)= f2i+12 P (n)2i+1P_(n)_1(for 0 i n - 1) defined by
equation (10.15) have the following properties:
(a) f0 = 11 and f1 = e;
P
(b) _fk = i+j=kfi fj;
(c) fflfi= 0 for all i > 0, in particular, ffla(i)= 0 for all i;
(d) a(i)Oa(j)= a(j)Oa(i);
i j
(e) fi* fj = i+jifi+j, provided i + j 2N;
(f) a(i)* a(i)= 0 for 0 i < n - 1;
(g) For all r, r*fk is the coefficient of tk in the formal identity
2N ____ i j OE _
r*f(t) = * {f(t)OiO[]} inP (n) P (n) [t] (t2N+1).|_|
i=0 * _____*
____ i j
Again, for i = n - 1, (f) is replaced by (10.14), now taken inP (n)* P_(n)_*
*1.
Finally, we prove in [Bo ] that equation (7.18) lifts in the obvious way.
____ i j
Lemma 10.17 In the Hopf ringP (n)* P_(n)_* for p = 2, we have
a(i)Oa(i)= b(i+1) for 0 i n - 1. (10.18)
Remark There is a case for writing e here as a(-1), so that the identity eOe *
*= b(0)
becomes a natural extension of (10.18).
11 Structure of the Hopf ring
i j
In this section, we present two descriptions of the Hopf ring P (n)* P_(n)_*
** [re-
____ i j
placed byP (n)* P_(n)_* if p = 2]: a clean concise description in terms of the *
*gener-
ators and relations developed in x10, and a concrete computational description *
*that
specifies exactly what the elements of the Hopf ring are. (This relies heavily *
*on the
technical work of Ravenel-Wilson [RW96 ], and in no way replaces it.)
i j
Theorem 11.1 (Ravenel-Wilson) The Hopf ring P (n)* P_(n)_* [which is replac*
*ed
____ i j
byP (n)* P_(n)_* if p = 2] over P (n)* has the O-generators:
i j
[vk] 2 P (n)0 P_(n)_-2(pk-1)for k n, defined by equation (10.4);
i j
e 2 P (n)1 P_(n)_1, defined by equation (10.6);
i j
b(j)= bpj2 P (n)2pj P_(n)_2for j 0, defined by equation (10.7);
i j
a(i)= api 2 P (n)2piP_(n)_1 for 0 i < n, defined by equation (10.12)
[replaced by (10.15) if p = 2];
- 39 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
subject to the relations [1]*p = 10, eOe = -b(0), the main relations (Rk) for k*
* > n
(see Definition 10.9), and the two variants (10.10) and (10.14) of (Rn) [also (*
*10.18)
if p = 2].
Allowable monomials For our second description of the Hopf ring, we reinterpret
the general monomial (8.1) as the O-monomial
eOfflOaOIObOJO[vK ]
(11.2)
= eOfflOaOi0(0)OaOi1(1)O.O.a.Oin-1(n-1)ObOj0(0)ObOj1(1)ObOj2(2)O.O.[.vn]*
*OknO[vn+1]Okn+1O. . .
(We adopt the usual convention (e. g. [RW77 ]) that for any element d with ffl*
*d = 0,
d O0= [1] - 10, so that d O0Od = d holds. We also set [vk]O0= [v0k] = [1].)
We define it to be allowable or Q-allowable exactly as in Definition 8.2.
A direct description of the allowable monomials is useful, to replace the i*
*ndirect-
ness of Definition 8.2. As in [RW96 ], 0 denotes the multi-index (1, 0, 0, . *
*.)..
Proposition 11.3 Any allowable O-monomial c in the Hopf ring can be written
uniquely in one of the standard forms
(a) c = aOIObOG+L O[vK ] if c does not involve e;
(11.4)
(b) c = eOaOIObOG+L- 0 O[vK ] if c involves e;
where the multi-index G is defined by
n Opn+1 Opq-1
bOG = bOp(dn)Ob(dn+1)O. .O.b(dq-1), (11.5)
L = (l0, l1, l2, . .)., and the indices satisfy
(i)q n;
(ii)0 dn dn+1 . . .dq-1;
(iii)0 lt< pr for all t < dr, for n r < q;
(11.6)
(iv)0 lt< pq for all t;
(v) kr = 0 (i. e. vK contains no factor vr) for all r < q;
(vi)In Case (b), dn = 0 or l0 > 0.
Conversely, any such monomial is allowable.
Proof If the allowable monomial c does not involve e, we choose each dr in tu*
*rn as
small as possible, so that (iii) holds; moreover, (iii) requires this choice of*
* dr. (If we
cannot even start, q = n, c = aOIObOLO[vK ], and (ii), (iii) and (v) become vac*
*uous.)
We continue as long as possible, until (iv) holds. In view of (8.3)(i), c does *
*not contain
[vr] for any r < q, and (v) holds.
If c has the form eOc0, we note that eOc = b(0)Oc0remains allowable, and ap*
*ply case
(a) to it. Here, we need (vi) so that 0 can be subtracted off.
_
Conversely, the monomials (11.4) are easily seen to be allowable. |_|
The algebra structure We recall that a simple system of generators of a graded
algebra A with multiplication * over a graded ring R of characteristic p is a s*
*et of
elements z1, z2, z3, . .s.uch that the finite products
z*M = z*m11* z*m22* z*m33* . .,. (11.7)
- 40 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
where 0 mr < p for each zr of even degree and mr = 0 or 1 for each zr of odd
degree, form a set of free R-module generators of A.
The following description is also essentially included in [RW96 , Thms. 1.*
*3, 1.4]
[except that for p = 2, (d) was not written out explicitly and contains the sur*
*prise
(iii)]. For I 6= (1, 1, . .,.1), ae(I) denotes the smallest t such that in-t = *
*0.
Theorem 11.8 (Ravenel-Wilson) Assume 0 < n < 1, and let k be any integer.
Then i j
____ i j
(a) The Hopf algebra P (n)* P_(n)_k [orP (n) *P_(n)_k if p = 2] has as a sim*
*ple
system of *-generators the set of all allowable O-monomials (11.2) (that lie in*
* it);
(b)iThe Q-allowablejO-monomials_formiajminimal set of algebra *-generators of
P (n)* P_(n)_*[orP (n)* P_(n)_* if p = 2];
i j
(c) For p odd, P (n)* P_(n)_k is the tensor product of the following subalge*
*bras,
one for each Q-allowable O-monomial (that lies in it):
(i) T Pae(I)(aOIObOJO[vK ]) for I 6= (1, 1, . .,.1);
(ii)P (aOIObOJO[vK ]) for I = (1, 1, . .,.1);
(iii)E(eOaOIObOJO[vK ]);
____ i j
(d) For p = 2,P (n)* P_(n)_k contains the following subalgebras, one for eac*
*h Q-
allowable O-monomial (that lies in it), and is additively (but not multiplicati*
*vely)
isomorphic to their tensor product:
(i) T Pae(I)(eOfflOaOIObOJO[vK ]) for I 6= (1, 1, . .,.1);
(ii) P (aOIObOJO[vK ]) for I = (1, 1, . .,.1);
(iii) T Pn+1(eOaOIObOJO[vK ]) for I = (1, 1, . .,.1).
Remark For p = 2, the quotient algebra
i j ____ i j
H* P_(n)_k; F2 ~= F2 P(n)*P (n)*P_(n)_k
is the tensor product of the subalgebras listed in (d), interpreted as F2-algeb*
*ras.
To complete this description, we need the structure maps *, O, _, ffl, and *
*O, which
are all (bi)linear. We know _, ffl, and O on each generator e, a(i), b(j)and [v*
*k]; then
the Hopf ring laws determine these operations in general.
Reduction to standard form We reprove part of Theorem 11.1 by showing that
we have enough relations to reduce any Hopf ring expression to a P (n)*-linear *
*com-
bination of *-products (11.7) of allowable O-monomials.
For O, we need to know how to O-multiply any two O-monomials (11.2); then t*
*he
distributive laws for (a* b)Oc and aO(b* c) [modified if p = 2] take care of ge*
*neral
*-monomials z*M as in (11.7). As the O-generators O-commute up to sign [even *
*for
p = 2], all we need is a reduction formula for each non-allowable O-monomial (1*
*1.2).
The relation eOe = -b(0)takes care of eO2. If p is odd, aO2(i)= 0 is automa*
*tic, by
Proposition 10.13(i). [If p = 2, we use aO2(i)= b(i+1)instead, from equation (1*
*0.18).]
- 41 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
For the disallowed monomials (i) and (ii) of (8.3), we use the same relatio*
*ns as
in Lemma 8.5, now working modulo *-decomposables as well. These use only the
relations (Rk) for k > n and (10.10), which implies (Rn).
For the *-product of two *-monomials (11.7), we shuffle the O-monomials into
the desired order (with the appropriate sign), and deal with excess *-powers of*
* any
O-monomial. [If p = 2, shuffling introduces extra terms, but the process quick*
*ly
terminates, because the *-commutator c * d - d * c of any two O-monomials is *-
central; see [Bo ] for details.]
The Frobenius operator To finish the reduction to standard form, we need a
formula for the Frobenius operator F c = c*p = c * c * . .*.c on each allowable
O-monomial c of even degree [or any degree if p = 2].
We start from the relation [1]*p = 10, which we rewrite as F ([1] - 10) = 0*
*. We
next reverse the identity [BJW95 , (15.13)] as
F (cObOJ) = (F c)ObO0,J, (11.9)
where 0, J denotes thePextended multi-index (0, j0, j1, j2, . .).. The proof u*
*sed only
the property _bk = i+j=kbi bj. Since ak has the same property when p is odd,
according to Proposition 10.13(b), we similarly have
F (aOI,0Oc) = aO0,IOF c
for any multi-index I = (i0, i1, i2, . .,.in-2). [For p = 2, Proposition 10.16 *
*delivers the
same result, and also
F (eOc) = a(0)OF c.]
For a(n-1), we rewrite the relation (10.14) as
F a(n-1)= vna(0)- a(0)ObON(0)O[vn].
Since applying -O[vK ] preserves *-multiplication, we immediately have
F (cO[vK ]) = (F c)O[vK ].
Combining these, we find the general formulae
F (aOI,0ObOJO[vK ]) = 0 (11.10)
and (with attention to the shuffles needed and the resulting signs)
F (aOI,1ObOJO[vK ]) = (-1)|I|+1aO1,IObON,JO[vnvK ] + (-1)|I|vnaO1,IObO0,JO[vK*
*(],11.11)
P
where |I| = rir.
[If p = 2, we need also the formulae involving e, which are
F (eOaOI,0ObOJO[vK ]) = 0 (11.12)
and
F (eOaOI,1ObOJO[vK ]) = aO0,IOb(1)ObON,JO[vnvK ] + vnaO0,IOb(1)ObO0,JO[vK(]*
*,11.13)
in which we make use of a(0)Oa(0)= b(1). For example,
n+1 O2n 2 ON O2n
F (eOa(n-1)ObON(0)) = bON+2(0)O[vn+1] + vnb(1)+ vnb(0)Ob(1)+ vn+1b(0)
after reduction to standard form, which recovers equation (7.21).]
- 42 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
A reduction formula There is a difficulty with equation (11.11) which obscures
the algebraic structure of the Hopf ring. Even in the simple case
F (a(n-1)ObOG) = -a(0)ObON,GO[vn] + vna(0)ObO0,G,
with G as in equation (11.5), the first term on the right is visibly not allowa*
*ble (unless
q = n, so that G = 0). What we need is a reduction formula for
n Opn+1 Opq-1
bO0,GO[vn] = bOp(dn+1)Ob(dn+1+1)O. .O.b(dq-1+1)O[vn],
which is essentially Lemma 3.8 of [RW96 ]. It involves the p-th O-power of bOG,
n+1 Opn+2 Opq
(bOG) Op= bOpG = bOp(dn)Ob(dn+1)O. .O.b(dq-1).
Lemma 11.14 Using only the main relations (Rk), the O-monomial bO0,GO[vn], w*
*ith
G as in equation (11.5), reduces to an allowable monomial by a formula of the f*
*orm
bO0,GO[vn] (-1)q-nbOpGO[vq] + . .,.
where the omitted terms do not involve any a(i)and either (i) have the form
bOJO[vk] with bOJ lexicographically higher than bOpG (see x8), (ii) lie in the *
*ideal
V = (vn, vn+1, . .)., (iii) have [v]-length at least 2, or (iv) are *-decompos*
*able.
We apply this to equation (11.11) [also (11.13) if p = 2].
Corollary 11.15 For the general allowable O-monomial (11.4)(a) without e, we
have
F (aOI,1ObOG+L O[vK ]) (-1)q-n+|I|+1aO1,IO{bON,LObOpGO[vqvK ] + .(.}..11*
*.16)
[If p = 2, we similarly obtain
F (eOaOI,1ObOG+L- 0 O[vK ]) aO0,IO{bON,LObO2GO[vqvK ] + . .}.(11.17)
_
from (11.4)(b).] |_|
The leading term on the right in equation (11.16) is always allowable: writ*
*ten in
0 K0 K0 K *
* OL0
standard form (11.4), it is aO1,IObOG+L O[v ], with the same G, v = vqv , a*
*nd b =
bON,LObO(p-1)G. Careful bookkeeping shows that, as the indices vary, it runs th*
*rough
all the Q-disallowed O-monomials of type (8.3)(iii) that are nevertheless allow*
*able,
once each. [Similarly, (11.17) accounts for types (iv) and (v).]
It follows that F never kills anything unexpected. Now we can read off part*
*s (c)
and (d) of Theorem 11.8.
n+1 Opn+2 Opr
Proof of Lemma For n r q, we set cr = bOp(dn)Ob(dn+1)O. .O.b(dr-1), so that
(conventionally) cn = bO0and cq = bOpG.
We show first that crO[vs] 0 whenever n s < r q, by induction on s.
k
We O-multiply (Rds+s) by cs; by (7.14), the k-th term is csObOp(ds+s-k)O[vk]. I*
*f k < s,
this term is neglected by induction. (If s = n, there are no such terms.) If *
*k > s,
we have ds + s - k < ds, and this term is lexicographically higher. If k = s, *
*we
s Ops+1-ps
have csObOp(ds)O[vs], which gives cs+1O[vs] 0 when we O-multiply by b(ds) ;*
* hence
crO[vs] 0 for any r > s, if we O-multiply by further factors.
- 43 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
s
Then we show that csObOp(ds+1)O[vs] -cs+1O[vs+1] for n s < q, from whic*
*h the
result follows by induction, starting from cn = bO0. We O-multiply (Rds+s+1) b*
*y cs.
k
The k-th term is csObOp(ds+s+1-k)O[vk], which we have just shown is negligible *
*if k < s.
If k > s + 1, we have ds + s + 1 - k < ds, and the term is lexicographically hi*
*gher.
_
The two remaining terms, with k = s and k = s + 1, are the desired terms. |_|
12 Effect on homotopy groups
Given an unstable operation r: P (n)k(-) ! P (n)m (-), where k, m > 0, cons*
*ider
the homomorphism of homotopy groups r*: kP (n)* ! m P (n)* (see diagram (3.9))
induced by the representing map r: P_(n)_k! P_(n)_m. By [BJW95 , Lemma 13.9], *
*it is
given on kv, where v 2 P (n)i, by the unstable analogue of equation (7.6), nam*
*ely
r* kv = m , (12.1)
where e2j= bOj(0)and e2j+1= eObOj(0).
We therefore seek more information on the relations in the Hopf ring.
The first higher-order relation We need the Hopf ring version for P (n) of Ben-
dersky's Lemma [Be86 , Thm. 6.2], which immediately implies Lemma 8.12.
i j
Lemma 12.2 For q n, we have in P (n)* P_(n)_g(n,q-1)+1the reduction formula
i j
eg(n,q)-1O[vq] vqeg(n,q-1)+1 mod IqP (n)* P_(n)_g(n,q-1)+1. (12.3)
[If p = 2, this is almost superseded by
eg(n,q)-2O[vq] vqeg(n,q-1)+ F (eg(n,q-1)-1Oa(n-1)) mod Iq, for q (n1+21.*
*].4)
Proof We establish (12.3) by induction on q. For q = n, it follows immediate*
*ly
from (10.10). ForPq > n, we return to the definition of the relation (Rq). We
expand [p](x) = K ~(K)vK xd(K), summing over multi-indices_K, with coefficien*
*ts
~(K) 2 Fp and exponents d(K); then if we write b(x) = 12 + b(x), (10.8) becomes
_`X K d(K)' ae _ Od(K) K oe*~(K)
12 + b ~(K)v x = * 12 + b(x) O[v ] . (12.5)
K K
We apply the suspension ehO-, where h = g(n, q-1) - 1, which kills 12 and m*
*ost
*-products and thus drastically simplifies (12.5) to
_` X K d(K)' X _ Od(K) K
ehOb ~(K)v x = ~(K)ehOb(x) O[v ].
K K
q
We take the coefficients of xp and work mod Iq. On the left, by (6.18), the o*
*nly
q
surviving term in [p](x) is vqxp , giving ehOvqb(0), the right side of (12.3). *
*On the right,
ehO[vk] 0 for all k n + 1, we have ehO[vq-1] F (eg(n,q-2)-1Oa(n-1)) by induction. Then
q-1 O2q-1 O2q-1
ehObO2(1)O[vq-1] b(1) OF (eg(n,q-2)-1Oa(n-1)) = F (b(0) Oeq(n,q-2)-1Oa(n*
*-1)),
_
as required, with the help of equation (11.9).] |_|
Proofs for x3 Now we can finish the proofs of two lemmas.
Proof of Lemma 3.8 For (a), by (12.1),
r* k(vnv) = m = m .
Since k + q > 0, we can use (10.10) to rewrite this as
r* k(vnv) = m = vnr* kv.
_
Part (b) is similar, with (12.3) in place of (10.10). |_|
Lemma 12.6 Let r: P_(n)_k! P_(n)_kbe any map, where k > g(n, m-1), and suppo*
*se
that on homotopy, r*: kP (n)* ! kP (n)* is given on the bottom class by r* k1*
* =
~ k1, where ~ 2 Fp. Then on any monomial vK in the elements vn, vn+1, . . . , v*
*m , r*
has the form X
r* kvK = ~ kvK + cL kvL (12.7)
L>K
with coefficients cL 2 Fp, where we order the multi-indices L = (ln, ln+1, . .)*
*.lexico-
graphically (as in x8).
Proof We use induction on the length of vK , starting from K = 0. If (12.7) h*
*olds
for kvK , Lemma 3.8(b) gives
X
r* k(vqvK ) ~ k(vqvK ) + cL k(vqvL) mod Iq.
L>K
If we assume (as we may) that vK contains no factors vt with t < q, all monomia*
*ls in
*
* _
Iq will be larger lexicographically than vqvK , and we have the result for vqvK*
* . |_|
Proof of Lemma 3.6 Take any map f: P_(n,_m)_k! P_(n,_m)_k, where g(n, m-1) <
k g(n, m). Suppose f* k1 = ~ k1. We apply Lemma 12.6 to the composite
___
P_(n)_k-ae(m)---!P_(n,_m)kf--!P_(n,_m)k`(m)----!P_(n)k
to deduce that ____ X
`(m) *f* kvK = ~ kvK + cL kvL
L>K
for any monomial vK in the generators vn, vn+1, . . . , vm . We apply ae(m)*, t*
*o see that
f* kvK has the same form (possibly with some terms vL deleted). It is now cle*
*ar
_
that if ~ 6= 0, f* is an isomorphism and f is a homotopy equivalence. |_|
- 45 -
J.M.Boardman, W.S.Wilson H-spaces and P (n)-cohomology
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Department of Mathematics, Johns Hopkins University
3400 N. Charles St., Baltimore, MD 21218-2689, U.S.A.
E-mail: boardman@math.jhu.edu, wsw@math.jhu.*
*edu
- 47 -
~~