UNSTABLE SPLITTINGS RELATED TO BROWN-PETERSON
COHOMOLOGY
J. MICHAEL BOARDMAN AND W. STEPHEN WILSON
Abstract.A new and relatively easy proof of various unstable splittings *
*as-
sociated with Brown-Peterson cohomology is presented.
1.Introduction
In [Wil75], unstable splittings were constructed for the spaces in the Omega
spectrum for Brown-Peterson cohomology, a cohomology theory with coefficient
ring BP *' Z(p)[v1; v2; : :;:vn; : :]:. This was done using the Postnikov decom*
*po-
sition and a multiple induction. The proof, from [Wil73] (again using Postnikov
systems), that these spaces had no torsion was essential in this proof. In [RW7*
*7 ],
the calculation of the homology of those spaces for BP was done as a Hopf ring,*
* a
great improvement, and this was used in [BJW95 ] to construct an unstable idem-
potent to get the splittings. Some of the splittings are not as H-spaces and so*
* the
full power of non-additive unstable operations was required. The splittings we*
*re
generalized in [BW ] to the spaces in the Omega spectrum for P (n), n > 0, a t*
*he-
ory with coefficient ring P (n)* ' BP *=In where In = (p; v1; v2; : :;:vn-1), a*
*fter
the calculation of the Hopf ring for these spaces in [RW96 ]. This calculation *
*was
done with the intent of getting an analogous splitting. The technique is again *
*to
construct an unstable idempotent to get the splittings.
Although the technique of constructing unstable idempotents is clearly the pr*
*oper
way to prove these results, it requires an immense amount of technical machinery
which cannot be accused of being easily accessible.
We wish to present a much more direct proof of these splittings which requires
none of the machinery of unstable operations. In fact, the proofs could be done*
* quite
easily if one could just insert a few short paragraphs into the papers [RW77 ] *
*and
[RW96 ]. Unfortunately that options is not open to us. If we essentially reprod*
*uce
those papers to insert what little extra is needed, then the proofs cease to be*
* "easy."
On the other hand, if we just create those paragraphs to be inserted, then the *
*result
remains obscure. We will try to walk a fine line between these two approaches. *
*Our
goal will be to write the necessary insertions in such a way that a rigorous pr*
*oof has
been accomplished when combined with the previous two papers but at the same
time discuss the results in enough depth so the reader should be convinced of t*
*he
result without having to consult the other papers.
First we need to establish some notation and state our results. We let P (0)
represent BP so BP is not an exceptional case. In fact it is both easier and qu*
*ite
different from the n > 0 case. There are also theories BP with homotopy
Z(p)[v1; : :;:vm ] and P (n; m) with homotopy BP *=In (I0 = (0), so BP =
____________
1991 Mathematics Subject Classification. Primary 55N22 ; Secondary 55N20 .
1
2 J. MICHAEL BOARDMAN AND W. STEPHEN WILSON
P (0; m)) which are associative ring spectra by [SY76 ]. If E is a spectrum we
denote the spaces in the Omega spectrum by {E_k}. Define g(n; m), for n m to
be 2(pn + pn+1 + . .+.pm ). We will show that for k g(n; m), P_(n;_m)_ksplits *
*off
of P_(n)_k. Precisely:
Theorem 1.1 ([Wil75], and later [BJW95 ], for n = 0 and [BW ] forFno>r0).
k g(n; m) there is an unstable splitting
Y
P_(n)_k' P_(n;_m)_k P_(n;_j)k+2(pj-1):
j>m
Remark 1.2.Once the bottom piece has been split off the rest of the splitting
follows easily. If k < g(n; m) then it follows that this is a splitting of H-sp*
*aces.
Remark 1.3.We do not recover the result of [BW ] for n > 0 and k = g(n; m) that
this is still a splitting as H-spaces. The fact that the homology splits off as*
* Hopf
algebras tells us nothing. If k > g(n; m) then it is easy to see from our appro*
*ach
that the homology of the smaller piece no longer splits off and so there is no *
*such
homotopy splitting as well. One of the major attractions about our approach is
that we don't have to worry in any way about the additivity of our splittings. *
*For
the n = 0, k = g(0; m) case where the splittings are not additive, this is a ma*
*jor
complication in the proof using unstable idempotents. Our approach is indiffere*
*nt
to such matters although it does show the non-additivity of this splitting beca*
*use
we see that the homology splitting cannot be as Hopf algebras.
Remark 1.4.Note that when n = m the small bottom piece of the splitting is just
a space in the spectrum for Morava K-theory.
Remark 1.5.Another major complication in [BW ] for the n > 0 case is the prime
2. The theories involved are not homotopy commutative ring spectra and so the
machinery for unstable idempotents must be extended and contorted to deal with
this special case. One of the benefits of our approach is once again that we do*
* not
need to worry about such things. The p = 2 proof is identical to the odd prime
proof and we need not be concerned whether any of the spectra are commutative
ring spectra or not. The p = 2 version of the theorem is very important.
A major motivation for the theorem, and even for a second or third proof, is
its important applications. First, it is easy to prove a generalization of Quil*
*len's
theorem from the splitting.
Theorem 1.6 (For n = 0, [Qui71], and, for n > 0, [BWFo].).r X a finite complex,
P (n)*(X) is generated by non-negative degree elements.
Proofs for the n = 0 case using the splitting appear in [Wil75] and [BJW95 ].
In this last case a more general, purely algebraic, version about unstable modu*
*les
is proven. Strickland has shown that Quillen's proof cannot be generalized to t*
*his
case.
Second, all of the results of [RWY98 ] are unstable, and the only unstable i*
*nput
is this generalized Quillen theorem. Everything depends on it. Thus, we feel it*
* is
important to have a relatively simple and accessible proof, especially one with*
* no
complications associated with the prime two or the non-additive splittings.
The authors are particularly indebted to the organizers of the 1998 Barcelona
Conference on Algebraic Topology and the Centre de Recerca Matematica for the
UNSTABLE SPLITTINGS RELATED TO BROWN-PETERSON COHOMOLOGY 3
conference and terrific work environment during the extended stay of the second
author. Thanks are also due to our previous coauthors, David Johnson and Douglas
Ravenel, for the work with them which led us to this paper.
2.Approach and simple parts of the proofs
In [RW77 ] and [RW96 ] the homology of the spaces in the Omega spectra were
calculated by induction on degree using the bar spectral sequence
TorH*(P(n)_*)(Z=(p); Z=(p)) ) H*+1(P_(n)_*+1):
The Hn+1 is there to indicate that we gain a degree in this inductive calculati*
*on.
What was not noticed in those two papers was that we could easily have simul-
taneously calculated the homology of the bottom piece which splits off. We would
just use the spectral sequence
TorH*(P(n;m)_*)(Z=(p); Z=(p)) ) H*+1(P_(n;_m)_*+1):
There is a map of spectra P (n) ! P (n; m) which induces a map on the spectral
sequences. The proof of the calculation of this spectral sequence is identical *
*up to
k = g(n; m) with the exception of one slightly modified definition. It is easy *
*to see
that the calculation cannot go one step higher.
Once the homology is calculated it is easy to see that the Atiyah-Hirzebruch
spectral sequence for the P (n) and P (n; m) homology of these spaces collapses*
* and
so they are free over the coefficient rings.
Corollary 2.1.For k g(n; m) the Atiyah-Hirzebruch spectral sequences
H*(P_(n;_m)_k; P (n)*) and H*(P_(n;_m)_k; P (n; m)*)
collapse and P (n)*(P_(n;_m)_k) is P (n)* free and P (n; m)*(P_(n;_m)_k) is P (*
*n; m)*
free.
This in turn allows us to calculate the cohomology theories as the dual and g*
*et:
Corollary 2.2.For k g(n; m), the map
P (n)*(P_(n;_m)_k) ! P (n; m)*(P_(n;_m)_k)
is surjective.
Proof of Theorem 1.1.To get our splitting we just note that the identity map of
P_(n;_m)_k, an element of P (n; m)k(P_(n;_m)_k), has a lift to a map of P_(n;_m*
*)_kto
P_(n)_k, an element of P (n)k(P_(n;_m)_k). That splits off the bottom piece. *
*The
other pieces are handled by taking the maps
P_(n;_j)k+2(pj-1)! P_(n)_k+2(pj-1)! P_(n)_k
where the first map is just the bottom piece splitting and the second map comes
from the stable map of vj. Using the H-space structure to add all of these maps
up we see we have a homotopy equivalence due to the obvious isomorphism on
homotopy groups.
This concludes the proof of the splitting from the calculation of the homolog*
*y_
and the collapsing of the Atiyah-Hirzebruch spectral sequence. |_*
*_|
We now want to give a proof of the Quillen theorem.
4 J. MICHAEL BOARDMAN AND W. STEPHEN WILSON
Proof of Theorem 1.6.This proof is exactly the same as the proof of the original
Quillen theorem given in [Wil75], but because it is of such major importance, a*
*nd
it is short, we reproduce it here. There are stable cofibrations
m -1) vm
(2.1) 2(p P (n; m) -! P (n; m) -! P (n; m - 1)
where P (n; n - 1) is the mod p Eilenberg-Mac Lane spectrum. These cofibrations
give rise to long exact sequences in cohomology theories. Given a negative degr*
*ee
element x 2 P (n)*(X) where X is a finite complex, we see that there is some
q n for which x maps to zero in P (n; q - 1)*(X) because mod p cohomology is
zero in negative degrees. Using the above exact squence there is an element yq 2
P (n; q)*(X) such that vqyq is the image of x in P (n; q)*(X). Because x has ne*
*gative
degree, the element yq, which is a map of X into the space P_(n;_q)|x|+2(pq-1),*
* is
in the range of the splitting Theorem 1.1 and so the element yq can be lifted to
P (n)*(X). We now look at the element x - vqyq. It will go to zero in P (n; q)**
*(X)
and we can iterate this process. It is a finite process since the splitting The*
*orem
1.1, combined with the finiteness of X, tells us that for some large m, PP(n)k(*
*X) is
the same as P (n; m)k(X) for a fixed k = |x|. At that point we have x = viyi,*
* a
finite sum. Thus any negative degree element is decomposable in this way and we
have proven the generalized Quillen theorem. (For the n = 0 case we let P (0;_0*
*) be
the Z(p)cohomology and we do not need the mod p cohomology.) |__|
3. The Homology
Let H*(-) be the standard mod p homology where p is the prime associated
with the spectrum P (n; m). Because P (n; m) is a ring spectrum there are maps
P_(n;_m)_ix P_(n;_m)_j-! P_(n;_m)_i+j
corresponding to cup product, in addition to the loop space product
P_(n;_m)_ix P_(n;_m)_i-! P_(n;_m)_i:
These induce pairings
O : H*(P_(n;_m)_i) H*(P_(n;_m)_j) ! H*(P_(n;_m)_i+j)
and
* : H*(P_(n;_m)_i) H*(P_(n;_m)_i) ! H*(P_(n;_m)_i):
Since H*(-) has a Kunneth isomorphism these pairings satisfy certain identiti*
*es
making H*(P_(n;_m)_*) into a Hopf ring, [RW77 ], i.e., a ring object in the cat*
*egory
of coalgebras.
There are special elements
e 2 P (n)1(P_(n;_m)_1);
a(i) 2 P (n)2pi(P_(n;_m)_1) for0 i < n;
[vi]2 P (n)0(P_(n;_m)_-2(pi-1))fori n; i > 0; and
b(i)2 P (n)2pi(P_(n;_m)_2) fori 0;
which have already been defined in [RW96 ] for P (n) and we get these by just
pushing them down using the map from P (n) to P (n; m) to get them first in
UNSTABLE SPLITTINGS RELATED TO BROWN-PETERSON COHOMOLOGY 5
P (n)*(P_(n;_m)_*) and then into P (n; m)*(P_(n;_m)_*). They then push down non-
trivially to H*(P_(n;_m)_*).
A basic property which we need and which comes out of the construction of the*
*se
elements, (this goes clear back to [Wil84]), is:
Proposition 3.1.The elements e, a(i), [vi], and b(i)are permanent cycles in the
Atiyah-Hirzebruch spectral sequence for P (n)*(-) and P (n; m)*(-).
Other facts proven about these elements are in [Wil84, Proposition 1.1] and w*
*ere
repeated again in [RW96 , Proposition 2.1, p. 1048] and are not repeated again *
*here.
Let
e"aI[vK ]bJ = e" O aOi0(0)O . .O.aOin-1(n-1)O [vknnvkn+1n+1.].O.bOj0(0)O*
* bOj1(1). . .
where " = 0 or 1, iq = 0 or 1, kq 0, and jq 0, (K and J finite), and if n = 0,
k0 = 0.
Definition 3.2.For n > 0 we say e"aI[vK ]bJ is nm-allowable if
J = pndn + pn+1dn+1+ . .+.pqdq+ J0
where d has a 1 in the dthplace and zeros elsewhere, dn dn+1 . . .dq and
J0 is non-negative implies kq = 0. In other words,
n +pn+1 +...+pq
[vq] O bp dn dn+1 dq
does not divide e"aI[vK ]bJ when dn dn+1 . . .dq, q < m. We will denote
the set of such (K; J) by Anm . If we eliminate the reference to m then we have*
* the
n-allowable of [RW96 ] Because we do not want to use [v0] in the n = 0 case we *
*set
A0m = A1m. There is still a difference between the allowable elements because I
is empty for n = 0 but not for n = 1.
We say e"aI[vK ]bJ is nm-plus allowable if e"aI[vK ]bJ+0 is nm-allowable. We
will denote the set of such (K; J) by A+nm. Note that A+nm Anm .
Define the shift operator s on J by
(3.2) bs(J)= bOj0(1)O bOj1(2)O . .:.
Theorem 3.3. Let H*(-) be the standard mod p homology with p the prime asso-
ciated with P (n; m).
For n > 0 and * g(n; m). H*(P_(n;_m)_*) is the same as H*(P_(n)_*) stated in
[RW96 , Theorem 1.3, p. 1045] except we replace the n and n-plus allowable with*
* nm
and nm-plus allowable. For p = 2 there is one more minor modification described
in the appendix to this paper.
For n = 0 and * < g(0; m),
O O
H*(P_(0;_m)_*) ' E(e[vK ]bJ) P ([vK ]bJ):
(K;J)2A0m (K;J)2A0m
For n = 0 and k = g(0; m), as a coalgebra, H*(P_(0;_m)_k) is the divided power
coalgebra O
([vK ]bJ+0 ):
(K;J)2A0m
The elements flpi([vI]bJ+0 ) represent [vI]bsi(J+0).
6 J. MICHAEL BOARDMAN AND W. STEPHEN WILSON
Remark 3.4.Of course we insist that one only uses the elements which actually
lie in thePappropriatePspaces. ThePelement e"aI[vKP]bJPis in Hs(P_(n;_m)_k) whe*
*re
s = " + 2piq+ 2pjqand k = " + iq + 2 jq - 2(pq - 1)kq.
Remark 3.5.For n = 0 and * < g(0; m), this is the same as in [RW77 ], replacing
allowable with 0m-allowable.
Proof of Corollary 2.1T.he Atiyah-Hirzebruch spectral sequence respects the two
products, O and *, and all elements in the P (n)*(P_(n;_m)_*) we are considering
are constructed using these two products from the basic elements e, a(i), [vi],*
* and
b(i)which are all permanent cycles by Proposition 3.1. Thus the spectral_sequen*
*ce
collapses. |__|
Primitives are calculated simultaneously as in [RW96 , Theorem 1.4, p. 1046]
and [RW77 ].
The p = 2 case deserves some discussion. The spectra P (n) and P (n; m) are n*
*ot
commutative ring spectra. However, the standard homology is still a commutative
Hopf ring, see the explanation in [Wil84, pages 1030-31]. There are no concerns
raised by this lack of commutativity in the collapse of the Atiyah-Hirzebruch s*
*pec-
tral sequence, the use of duality to compute the cohomologies or their applicat*
*ion
to get the splittings. The lack of commutativity could make things very bad for
some applications but it doesn't interfere in the slightest with what we are do*
*ing.
It does make the proof of the splitting in [BW ] much harder.
The proof of our theorem relies on being able to identify elements in the bar
spectral sequence, compute differentials and solve multiplicative extension pro*
*b-
lems, all using Hopf ring techniques. The n = 0 case has no differentials but d*
*oes
have extension problems.
Let Q stand for the indecomposables.
Theorem 3.6. In QH*(P_(n;_m)_k), k g(n; m), any e"aI[vK ]bJ can be written in
terms of nm-allowable elements.
Proof. The construction and proof of an algorithm for the reduction of non-
allowable elements is done on pages 273-275 of [RW77 ]. The proof applies with
only notational modification to the case of nm-allowable when I = 0. We can the*
*n_
circle multiply by aI to get our result. |__|
The homology and primitives are calculated simultaneously by induction on de-
gree in the bar spectral sequence. Recall that for a loop space X with classify*
*ing
space BX the bar spectral sequence converges to H*(BX), and its E2-term is
TorH*(X)*;*(Z=(p); Z=(p)):
When BX is also a loop space, we have a spectral sequence of Hopf algebras.
The E2 term of the bar spectral sequence for n > 0 is calculated inductively *
*from
Theorem 3.3 just as in [RW96 , Lemma 3.6, p. 1056]. For n = 0 the calculation is
trivial as in [RW77 ]. Both cases must use the modified definition of allowable.
The complete behavior of the bar spectral sequence, using the modified defini*
*tion
of allowable, is given in [RW96 , Theorem 3.7, p. 1057] for n > 0 and in [RW77 *
*] for
n = 0. For n > 0 it is a gruesome description of all differentials and identifi*
*cation
of elements in terms of the Hopf ring. There are no differentials for n = 0 and*
* the
identifications are much easier as well.
UNSTABLE SPLITTINGS RELATED TO BROWN-PETERSON COHOMOLOGY 7
This does not complete the calculation of the homology but only the E1 term
of the spectral sequence. Extension problems must be solved. However, first, si*
*nce
we claim that the proofs of all parts of the calculation are exactly the same a*
*s for
the spaces in the Omega spectra for P (n) and BP as in [RW96 ] and [RW77 ], some
explanation is clearly needed in order to explain the differences between the t*
*wo
cases and to see why we cannot proceed up the Omega spectrum with P (n; m). That
difference comes about in the calculation of the differentials. For n = 0 there*
* are no
differentials so we do not see any difference at this step. For n > 0 the diffe*
*rentials
are not really calculated but inferred. Certain elements are shown to disappear*
* and
it is proven that they must be targets of differentials. They are counted and s*
*hown
to be in one-to-one correspondence with possible sources of differentials. Thu*
*s,
all possible sources must kill all necessary targets. The difference comes in *
*the
counting process. We use the following lemma and the difference is found in the
proof, which we include.
Lemma 3.7. Let n > 0. In H*P_(n;_m)_* 0 case that matters here, is done for differentials and the targets all hav*
*e an
e with them which throws this problem up to the k = g(n; m) + 1 space. We are
not working in this range so this doesn't affect us. It does tell us that the s*
*plitting
cannot be delooped though since it does say we cannot have as many differentials
on our smaller space as we would need to get the size of the homology down to
where it splits off. That is, all of our necessary targets are not there in the*
* next
space so some possible sources will survive. They do not survive in the space f*
*or
P (n) though, so the homology cannot split off.
The final problem which arises is the solving of the extension problems to gi*
*ve
us the proper homology. Again, the proofs are the same for the P (n; m) and P (*
*n)
cases. The counting argument of the previous proof is used again in this proof.
Here we need to solve various extension problems. First, we show that certain
elements can not be generators and then we show that the only thing that can
prevent them from being generators is if they are p-th powers. We then show that
they are in one-to-one correspondence with the only elements which could possib*
*ly
8 J. MICHAEL BOARDMAN AND W. STEPHEN WILSON
have non-trivial p-th powers. We use the same counting Lemma as in the previous
proof. This time, in the n > 0 case, we must always have an a(i)involved in the
p-th powers which again throws it up to the g(n; m) + 1 space before we see the
creation of an unwanted [vm+1 ]. However, in the n = 0 case, we see our p-th po*
*wer
extensions can, and do, actually occur in the g(0; m) space. Our result, in thi*
*s case,
is only correct as coalgebras and in the small space the homology is not a poly*
*nomial
agebra and so cannot split off of the homology of the larger space as Hopf alge*
*bras,
thus preventing the splitting from being as H-spaces. This completes the proof.
4.Appendix: p = 2
This paper requires the results of [RW96 ]. However, in that paper the results
were not stated explictly for p = 2. Because p = 2 is such an important part of
the contribution of this paper, we must rectify that enough to do the p = 2 case
here. The lack of precision with p = 2 in [RW96 ] originates in a similar vague*
*ness
in [Wil84]. First, we will straighten out the situation in [Wil84] and then we *
*will
do the same for [RW96 ].
The key to the solution is mentioned in the p = 2 comments in [Wil84, page
1030], namely, the element e must be included in the coproduct of the elements
ai. In particular, the Verschiebung is evaluatednas V (a(0)) = e. Using the mod*
* 2
homology formula, a*2(n-1)= a(0)O [vn] O b2(-10), we can compute
n-1
(e O a(n-1))*2= a(0)O (a(n-1))*2= a(0)O a(0)O [vn] O b2(0):
We note that
V (a(0)O a(0)) = V (a(0)) O V (a(0)) = e O e = b1 = b(0):
Since V (b(1)) = b(0)also and b(1)is the only element in degree 4 ofnthis space*
*, we
must have a(0)O a(0)= b(1). Thus, we have (e O a(n-1))*2= [vn] O b2(-10)O b(1).*
* For
p odd, all elements with e in them were exterior. However, for p = 2, we need to
look at all elements containing e O a(n-1)in both the cases we are considering.
For odd primes, we recall the result of [Wil84, Theorem 1] as
O I J O I J O I J
H*K(n)_*' E(a b O e1) TPae(I)(a b ) P(a b )
j0 0 is the smallest k with in-k = 0.
For p = 2 we must modify this to take into account the additional nontrivial
squares which we have already identified. In this case our description is preci*
*se.
Theorem 4.1. For p = 2, with the above constraints, H*K(n)_*'
O I J O I J
E(a b O e1) TPae(s(I-n-1))+1(a b e1)
j0< 2n - 1 j0< 2n - 1
in-1= 0 in-1= 1
O I J O I J
TPae(I)(a b ) P(a b )
I 6= I(1) I = I(1)
ifi0= 1; j0< 2n - 1
thenj0< 2n - 1
ifi0= 0 andj0= 2n - 1
thenj1= 0
UNSTABLE SPLITTINGS RELATED TO BROWN-PETERSON COHOMOLOGY 9
The essentials of the proof remain unchanged. We have taken the exterior gen-
erators which should be truncated polynomial generators and we have taken away
the generators which are their squares. The size remains the same in either des*
*crip-
tion. This is significantly easier than our next case because here we can solve*
* our
extension problems precisely.
It is tedius to reproduce the results of [RW96 ] here and then modify them sl*
*ightly.
We will keep the flavor of the rest of the paper and only produce the modificat*
*ions.
Theorem 4.2. For p = 2, the standard mod 2 homology, H*(P_(n)_*), fits in a
short exact sequence of Hopf algebras with the associated graded algebra being *
*given
by [RW96 , Theorem 1.3] (the odd prime answer). The quotient Hopf algebra is
just the exterior algebra on generators eaI[vK ]bJ as in [RW96 , Theorem 1.3] w*
*ith
in-1 = 1. These elements, in the actual algebra, all have nontrivial squares wh*
*ich
are contained in the set of generators of the subalgebra given by aI[vK ]bJ wit*
*h i0 = 0
and (K; J) 2 An - A+n.
Proof.In the Morava K-theory case we could evaluate the necessary squares di-
rectly. Here we cannot. It is essential that we know the squares are all nontri*
*vial
and linearly independent but it is not obvious how to do that directly. However,
the proof in [RW96 ] need only be modified slightly. In particular, we can stil*
*l work
in the same bar spectral sequence with the same elements. We must consider the
elements aI[vK ]bJ with i0 = 0 and (K; J) 2 An - A+n. After double suspension to
aI[vK ]bJ+0 we know this is zero mod * since (K; J) 62 A+n. However, it cannot
be a square because it has no a(0)in it and its degree is a multiple of 4. (If *
*it was
the square of an elements with e in it, thus making the a(0)unnecessary, then it
would have to have degree 2 mod 4). All differential targets must be odd degree*
* so
our double suspended element must be zero which implies something happened to
it in the previous spectral squence as eaI[vK ]bJ. Since it is odd degree it ca*
*nnot
be a square so it either is the target of a differential or it was already zero*
*. Both
happen. The counting argument of [RW96 , page 1061-2] pairs these up with the
potential source of differentials. For p = 2, the count, as given, uses fl2(oee*
*aI[vK ]bJ)
when in-1 = 1 and (K; J) 2 A+n. However, at p = 2, this would have to be a d1
differential which does not exist. We know that the elements which should be ta*
*r-
gets must be zero so the ones associated with these fl2 must have already been *
*zero.
These are in 1-1 correspondence with our fl2, or our eaI[vK ]bJ when in-1 = 1 a*
*nd
(K; J) 2 A+n. These are precisely the elements we wish to have non-trivial squa*
*res!
Thus, the only solution to our problem is that they have non-trivial sauares, l*
*in-
early independent, among the aI[vK ]bJ with i0 = 0 and (K; J) 2 An-A+n. Because
they are squares, they never show up as oeaI[vK ]bJ in the next spectral sequen*
*ce
so they do not have to be killed there. Likewise, because eaI[vK ]bJ is not ext*
*erior,
the fl2 we worried about in the spectral sequence does not exist so those unwan*
*ted
elements go away as well. This concludes the the discussion of the differences_*
*for
p = 2. |__|
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E-mail address: jmb@math.jhu.edu, wsw@math.jhu.edu