HOMOTOPY CLASSIFICATION OF SPACES
WITH INTERESTING COHOMOLOGY
AND A CONJECTURE OF COOKE,
PART I
J. Aguade, C. Broto and D. Notbohm
1. Introduction. The title of this paper is reminiscent of the title of one of *
*the last papers
by George Cooke ([7]). In that paper, Cooke observes that if X is a p-complete *
*loop space
then there is an action of [S1^p; S1^p] ~=^Zpon X. In particular, there is an a*
*ction of the p-1
roots of unity on X and by taking the quotients of appropriate loop spaces by t*
*his action
he obtains spaces with "interesting" cohomology, i.e. spaces whose cohomology a*
*lgebras
have quite few generators and relations and whose attaching maps represent inte*
*resting
elements in the stable homotopy of spheres. By applying this technique to S3<3*
*>, the
3-connective covering of S3, and to the fibre of the map S3 ! K(Z; 3) of degree*
* p, Cooke
constructs spaces realizing the cohomology algebras (subscripts denote degrees)
(1) Fp[x2n] E(fix2n);
where n is any divisor of p(p - 1), fi is the Bockstein homomorphism and E(yk) *
*denotes
an exterior algebra on one generator yk of degree k. This method was generalize*
*d in [5] to
construct spaces whose mod p cohomology has the form P E where P is a polynomi*
*al
algebra and E is an exterior algebra on the Bocksteins of the generators of P .*
* Cooke ends
his paper by saying "I expect that the condition n|p(p - 1) is necessary as wel*
*l" and this
is the Cooke conjecture that we mention in the title of this paper. The conjec*
*ture was
proved to hold true in some particular cases in [2] where cohomology algebras o*
*f the form
Fp[x] E(y) where studied by completely different methods to those used in the *
*present
paper.
Our purpose is to develop a study of spaces whose mod p cohomology has the f*
*orm
(1). In this part I of our work we consider the case of p odd and we prove the*
* Cooke
conjecture in full generality but we go further than that for we obtain a class*
*ification up to
p-completion of all homotopy types with mod p cohomology of the form (1) above.*
* When
p = 2 both the results and the techniques involved in the proofs are significan*
*tly different
and deserve a separate discusion which we plan to work out in part II of this w*
*ork ([4]).
In particular, the p = 2 version of the Cooke conjecture (which was certainly s*
*tated with
only the case of p odd in mind) turns out to be wrong and additional fascinatin*
*g families
of spaces with "interesting cohomology" appear.
_____________
J. Aguade and C. Broto are partially supported by DGICYT grant PB89-0321.
Typeset by AM S-*
*TEX
1
2 J. AGUADE, C. BROTO AND D. NOTBOHM
We present in this introduction a rough overview of the main results of the *
*paper. We
start with a classification of unstable algebras over the Steenrod algebra of t*
*he form (1).
We see that there are exactly two families of such algebras which we call Bi;ra*
*nd Ar for
i 0 and r dividing p - 1. As graded algebras, we have (subscripts denote degre*
*es)
Bi;r~=Fp[x2pir] E(y2pir+1)
and the Steenrod algebra action is determined by fi(x) = y and P pi(y) = (r - 1*
*)xsy. Ar
is isomorphic to B0;ras graded algebras but the action of the Steenrod algebra *
*is different
in B0;rand Ar for in Ar we have the relation P 1(y) = rxsy. The algebras reali*
*zed by
Cooke in [7] are B0;rand B1;rwhile the algebras Ar seem to have remained unnoti*
*ced
although their study will be fundamental in our classification of spaces realiz*
*ing Bi;r. In
this context, the conjecture of Cooke is stated as follows:
Theorem A. If H*(X; Fp) ~=Bi;ras algebras over the Steenrod algebra, then i 1.
It is a natural question to ask about the realizability of the algebras Ar. *
* It turns
out that all algebras Ar are realizable as mod p cohomology of some appropriate*
* spaces.
More in general, for any k 0 we introduce the notation A(k)rto denote a cohomo*
*logy
algebra which looks like Ar except for the fact that the relation fi(x) = y is *
*replaced by the
relation fi(k+1)(x) = y where fi(k+1)denotes the Bockstein homomorphism of orde*
*r k+1. In
particular, A(0)1= A1. Of course, fi(k+1)is not a Steenrod operation for k > 0 *
*and so A(k)r
0)
is the same as A(kr as algebras over the Steenrod algebra for any k; k0> 0. Nev*
*ertheless,
it makes sense to say that the mod p cohomology of some space X is isomorphic t*
*o A(k)r.
Theorem B. For k 0 and r|(p - 1) there is a p-complete space Xk(r) such that
H*(Xk(r); Fp) ~=A(k)r.
The spaces in theorem B are constructed by first taking the quotient of (BS1*
*)bpby some
appropriate action of the p-adic integers and then killing the one dimensional *
*skeleton.
Having established which algebras Bi;r, Ar are realizable, we consider the p*
*roblem of
classifying up to p-completion all homotopy types which realize these algebras.*
* In the case
of the algebras Ar we obtain that the spaces of part (1) of theorem B form a co*
*mplete list
of p-complete homotopy types realizing the algebras A(k)r.
Theorem C. Let H*(X; Fp) ~=A(k)r. Then X^p' Xk(r).
It is interesting to note that in proving this theorem we face the problem o*
*f comput-
ing the mod p cohomology of some component of map (BZ=p; X) in a case in which *
*the
appropriate T functor does not vanish in degree 1.
Finally, we consider the problem of classifying up to p-completion all homot*
*opy types
realizing Bi;r. Because of theorem A, we only need to deal with the cases of B*
*0;rand
B1;r. We obtain the amazing result that for each of these algebras there are i*
*nfinitely
many different p-complete spaces realizing it.
HOMOTOPY CLASSIFICATION 3
Theorem D. Let r|(p - 1) . There are spaces Yk;rfor 0 k 1 and Zk;rfor 0 < k 1
such that
H*(Yk;r; Fp)~=B1;r
H*(Zk;r; Fp)~=B0;r
All these spaces are p-complete and have different homotopy type.
Here Y1;r and Z1;r are the p-completions of the spaces constructed by Cooke *
*([7]). In
particular, Y1;1 = S3<3>bpand theorem D shows that there is an infinite family *
*of "fake"
S3<3>, i.e. spaces with the same mod p cohomology as S3<3> but not homotopy equ*
*ivalent
to S3<3> even after p-completion. Among these spaces the true S3<3> is distingu*
*ished by
being the only one which can carry an H-space structure.
Our next result shows that the spaces of theorem D form a complete list of p*
*-complete
homotopy types realizing the algebras B0;rand B1;r.
Theorem E. (1) If H*(X; Fp) ~=B1;rthen there exists 0 k 1 such that ^Xp' Yk;*
*r.
(2) If H*(X; Fp) ~=B0;rthen there exists 0 < k 1 such that X^p' Zk;r.
Finally we study suspensions of all the spaces constructed. It turns out tha*
*t even after
an l-fold suspension all the "fake" spaces are not homotopy equivalent to the g*
*enuine ones;
i.e. the spaces which were constructed out of S3.
Theorem F. If k 6= 1, for all r|(p - 1) and for all 0 l < 1, the l-fold suspen*
*sions
lYk;rand lY1;r are not homotopy equivalent.
The analogous statement is true for the spaces Zk;r.
The method used to prove most of the theorems stated above is based on the s*
*tudy
of the mapping spaces map (BZ=p; X), where X is a space whose cohomology is ass*
*umed
to be of the form Fp[x] E(y). Here the techniques developed by Lannes ([17]) *
*play a
fundamental role.
In order to show in a simplified way the main ideas in the proofs of theorem*
*s A to E
above, we present now a rough description of the homotopy classification of spa*
*ces with
the same cohomology as S3<3>. This will also illustrate where the fake S3<3> co*
*me from.
Imagine we have a p-complete space X with the same mod p cohomology as S3<3>. T*
*ake
Y = map (BZ=p; X)f to be an appropriate component of the space of maps from BZ=*
*p to
X and compute, using the T functor, the mod p cohomology of Y . It turns out th*
*at Y is
homotopy equivalent to X but the gain from X to Y is that Y exhibits a greater *
*symmetry
than X for Y belongs to a principal fibration
BZ=p ! Y ! Y (1):
The cohomology of Y (1) has the form Fp[x2] E(fi(x2)) hence either H*(Y (1); F*
*p) ~=B0;1
or H*(Y (1); Fp) ~=A1 and it turns out that both cases are possible. Hence, we *
*already have
two possibilities for X: the true S3<3> obtained by taking Y (1) to be Cooke's *
*realization
of B0;1and a fake one obtained by taking Y (1) equal to the space X0(1) of theo*
*rem B. If
4 J. AGUADE, C. BROTO AND D. NOTBOHM
H*(Y (1); Fp) ~=B0;1then we can apply the same technique again and obtain a pri*
*ncipal
fibration
BZ=p ! Y (1) ! Y (2)
with again two possibilities for Y (2). At the end we obtain either an infinite*
* sequence
X ! Y ! Y (1) ! Y (2) ! : :!:Y (j) ! : : :
with all spaces having mod p cohomology of type B(k)i;ror a finite sequence
X ! Y ! Y (1) ! Y (2) ! : :!:Y (j);
where the last space has mod p cohomology of type A(k)1, stopping the inductive*
* process
because map (BZ=p; Y (j))f will not be homotopy equivalent to Y (j). The first *
*case forces
X ' S3<3>bpand in the second one we obtain an infinite family of fake S3<3>. Mo*
*reover,
the uniqueness of realizations of the algebras A(k)ryields the homotopy uniquen*
*ess of each
of the fake spaces.
The paper is organized as follows. Section 2 deals with the algebraic proble*
*m of classi-
fying unstable algebras over the Steenrod algebra of the form Fp[x] E(y) with *
*fi(x) = y.
There we introduce the algebras Bi;rand Ar. In section 3 we compute the T func*
*tor
applied to these algebras, a computation that will be crucial for the rest of t*
*he paper.
In section 4 we prove the Cooke conjecture, i.e. theorem A (cf. theorem 4.3). *
*Section 5
is devoted to the construction of spaces whose cohomology is of the form A(k)r.*
* Here we
prove theorem B (cf. theorem 5.5). In section 6 we obtain the homotopy classifi*
*cation of
the spaces of section 5, proving theorem C (cf. theorem 6.1). Section 7 deals *
*with the
construction of spaces realizing Bi;rand in particular we obtain the family of *
*fake S3<3>
and we prove theorem D (cf. propositions 7.1 and 7.7 and corollary 7.6). In sec*
*tion 8 we
show that there are no more p-complete spaces realizing the algebras Bi;rbeside*
* those
constructed in section 7, by proving theorem E (cf. theorem 8.2). In section 9 *
*we study
suspensions of all the constructed spaces and prove theorem F (c.f. corollary 9*
*.13 and
corollary 9.15) using the localization functor of [9]. A final section 10 conta*
*ins some tables
which may help the reader through the rather intricated notation we use to deno*
*te the
spaces we are dealing with and their cohomology algebras.
The first and second author would like to thank the Sonderforschungsbereich *
*170 in
G"ottingen and specially L. Smith for the kind hospitality which made possible *
*the joint
work which has lead to the present paper. The third one would like to thank the*
* Centre
de Recerca Matematica in Barcelona for bringing together the authors again. All*
* of us are
grateful to Fred Cohen for many helpful discussions.
Warning. Throughout this paper p denotes an odd prime.
2. Some unstable algebras over the Steenrod algebra. Through this section we
say that A is a P E-algebra if A is a commutative graded Fp-algebra which is th*
*e tensor
product of a polynomial algebra on a generator of degree 2n and an exterior alg*
*ebra on
a generator of degree m. We say that A has type (2n; m). We will usually call*
* x one
polynomial generator and y one exterior generator.
HOMOTOPY CLASSIFICATION 5
Our first example of an unstable algebra over the Steenrod algebra is a P E-*
*algebra A
of type (2; 3). We define an unstable action of the mod p Steenrod algebra over*
* A by the
Cartan formula and the identities
P 1x = xp ; P ix = 0; i > 1 ; fix = y;
P 1y = xp-1y ; P iy = 0; i > 1 ; fiy = 0:
These formulas certainly define an unstable action of A over A, where A is th*
*e free as-
sociative algebra generated by P i, i > 0, and fi. In order to see that this ac*
*tion factors
through the Steenrod algebra A we have to check that the Adem relations hold in*
* A. This
could be done directly using the techniques in [24] but it follows also from th*
*e following
alternative description of A as a module over the Steenrod algebra.
Let H be the mod p cohomology of BZ=p and let P H be the even dimensional
subalgebra. P is a polynomial algebra on one generator v in degree 2. Let us de*
*note by
P + the submodule of P formed by the elements of positive degree. Consider the *
*diagram
in U (the category of unstable modules over the Steenrod algebra):
ss H OE
H -! _____P-+P
where ss is the natural projection and OE is the homomorphism given by
OE(vn) = noe(uvn-1 )
where u is a one dimensional generator in H such that fiu = v and oe denotes su*
*spension.
One can easily check that OE is an A-homomorphism. Actually, OE is the composit*
*ion
k1 2 2 j H
P -! P P --! Fp P ~= P!- _____P;+
where is the diagonal, k is the projection and j is an inclusion sending oe2vn*
* to oe(uvn).
Then if A is the pull back of the above diagram, A is an unstable module over t*
*he Steenrod
algebra and a straightforward computation shows that A ~=A as A -modules. This *
*shows
that the Adem relations hold true in A since they hold true in A.
If is a unit in Fpthen the map x 7! x induces an algebra automorphism of A *
*which
commutes with the Steenrod algebra action. Hence, for any r dividing p - 1 we h*
*ave an
action of the cyclic group of order r on A and the algebra of invariants of thi*
*s action is also
an unstable algebra over the Steenrod algebra. We call this algebra Ar. It is a*
* P E-algebra
of type (2r; 2r + 1) and the action of the Steenrod algebra is determined by:
P 1X= rXs+1;
fiX= Y;
P 1Y= rXsY;
where s = (p - 1)=r.
6 J. AGUADE, C. BROTO AND D. NOTBOHM
Let i 0. Our second example of an unstable algebra over the Steenrod algebr*
*a is an
algebra B which is a P E-algebra of type (2pi; 2pi+ 1). We define an unstable a*
*ction of
the Steenrod algebra over B by the Cartan formula and the identities
i p k i
P px = x ; P x = 0; k 6= 0; p ;
P ky = 0 for anyk > 0;
fix= y ; fiy = 0:
As before, a direct calculation as in [24] would check that the Adem relations *
*hold, but we
will instead use an alternative description of B as an unstable module over the*
* Steenrodi
algebra. Let H be as before and let P (i) be the subalgebra of H generated by v*
*p . Let
J(2) be the reduced mod p cohomology of S1 [p e2. Consider the diagram in U
i-1 ss2pi OE
2p J(2) P (i) -! P (i)- P (i)
where the map ss is the natural projection and OE is given by the composition
2pi 2pi
P (i) -! P (i) P (i)!- Fp P (i) ~= P (i);
or, equivalently, by the formula
i 2pi (n-1)pi
OE(vp ) = noe (v ):
Then if B is the pull back of this diagram, B is an unstable module over the *
*Steenrod
algebra such that B ~=B as A -modules. As before, this shows that B is an uns*
*table
algebra over the Steenrod algebra.
If is a unit in Fpthen the map x 7! x induces an algebra automorphism of B *
*which
commutes with the Steenrod algebra action. This produces, in the same way as b*
*efore,
algebras Bi;rfor any i 0 and any r dividing p - 1, with generators X and Y in *
*degrees
2pir and 2pir + 1, respectively, such that
i s+1 pj
P pX = rX ; P X = 0; j 6= i;
fiX = Y;
i s pj
P pY = (r - 1)X Y; P Y = 0; j 6= i:
Notice that Ar and B0;rare isomorphic as graded algebras, both being the ten*
*sor
product of a polynomial algebra on one generator x in degree 2 and an exterior *
*algebra
on one generator y in degree 3. Moreover, the relation fix = y holds in both a*
*lgebras.
However, Ar and B0;rare not isomorphic as algebras over the Steenrod algebra, a*
*s one
can easily check.
By construction, we see that if t divides r then Ar is a subalgebra of At an*
*d Bi;ris a
subalgebra of Bi;t. There are no further inclusions between these unstable alge*
*bras over
the Steenrod algebra.
The next theorem proves that there are no more examples of P E-algebras whic*
*h are
unstable algebras over the Steenrod algebra and such that fix = y.
HOMOTOPY CLASSIFICATION 7
Theorem 2.1. Let A be a P E-algebra which is an unstable algebra over the Steen*
*rod
algebra and such that fix = y. Then A is isomorphic, as an algebra over the St*
*eenrod
algebra, to one of the algebras Ar, Bi;rconstructed above.
Proof. Notice that the ideal generated by y in A is closed under the action of *
*the Steenrod
algebra. Hence A= ~=Fp[x] should be an unstable algebra over the Steenrod a*
*lgebra.
It is well known (cf. [25]) that this implies that if A is of type (2n; 2n + 1)*
* then n = pir
for some i 0 and some r dividing p - 1. Put s = (p - 1)=r. Then we can choos*
*e the
generator x such that i
P px = rxs+1:
This well known fact admits a tedious elementary proof using the Adem relations*
* and is
also a trivial consequence of the Adams-Wilkerson embedding theorem ([1]). If i*
* > 0 we
can use the Adem relation
i-1 pi pi
P 1fiP p = -fiP + P fi
to deduce i
P py = (r - 1)xsy:
By dimensional reasons and unstability, P pjx = 0 = P pjy for any j 6= i. Henc*
*e A is
isomorphic to Bi;r.
In the case i = 0 if we write P 1y = xsy then the Adem relation
2P 1fiP 1= fiP 1P 1+ P 1P 1fi
gives the following degree 2 equation for :
2 + (1 - 2r) + r(r - 1) = 0
whose roots are = r; r - 1. In the first case A is isomorphic to Ar and in the*
* second one
it is isomorphic to Bi;r.
Some of these P E-algebras appear as the mod p cohomology of some spaces. Le*
*t S3<3>
denote the 3-connective covering of S3, i. e. the fibre of the degree one map S*
*3!- K(Z; 3).
Then one can easily deduce from the spectral sequence of the fibration
K(Z; 2) ! S3<3> ! S3
and theorem 2.1 that
H*(S3<3>; Fp) ~=B1;1
as algebras over the Steenrod algebra. Moreover, since S3<3> is a loop space, *
*the p-
completion of S3<3> carries an action of the cyclic group of order p - 1 and by*
* taking the
homotopy quotient of S3<3>bpby the restriction of this action to the cyclic gro*
*up of order
r, for any r dividing p - 1, we obtain a space Xr such that H*(Xr; Fp) ~=B1;r. *
*(See [7] for
further details on this construction.)
8 J. AGUADE, C. BROTO AND D. NOTBOHM
If Y is the fibre of the map S3 ! K(Z; 3) of degree p then
H*(Y ; Fp) ~=B0;1
as algebras over the Steenrod algebra. Again, the p-completion of Y carries an *
*action of
the cyclic group of order p - 1 and this produces p-complete spaces Yr for any *
*r dividing
p - 1 such that H*(Yr; Fp) ~=B0;ras algebras over the Steenrod algebra. (See al*
*so [7] for
details.)
Finally, in section 5 we will prove that all P E-algebras Ar are realizable.
Even if we are only interested in the P E-algebras with fi(x) = y it will be*
* necessary
to consider P E-algebras where the Bockstein homomorphism acts trivially. They*
* will
indeed play an important role in the forthcoming sections. We denote by A01the*
* P E-
algebra of type (2; 3) with an unstable action of the Steenrod algebra given by*
* fi = 0 and
P 1(y) = xp-1y. Let A0rfor r dividing p-1 be the algebra of invariants of A01by*
* the action
of the cyclic group of order r which sends x to x and y to y for an r-th root *
*of unity.
A0ris a P E-algebra of type (2r; 2r + 1) with an unstable action of the Steenro*
*d algebra.
Let B0i;1be the same graded algebra Bi;1but with the Steenrod algebra action gi*
*ven by
fi = 0 and P t(y) = 0 for t 0. Let B0i;rfor r dividing p - 1 be the algebra of*
* invariants of
B0i;1under the action of the cyclic group of order r which sends x to x and y t*
*o y for
an r-th root of unity.
We denote by C1 a P E-algebra of type (2; 1) with an unstable action of the *
*Steenrod
algebra given by fi(x) = xz where x denotes a 2 dimensional generator and z den*
*otes a
one dimensional generator. Notice that A1 is isomorphic to a subalgebra of C1. *
*The cyclic
group of order r for r dividing p - 1 acts on C1 leaving z and xr invariants. W*
*e denote by
Cr the algebra of invariants, which is a P E-algebra of type (2r; 1) with an un*
*stable action
of the Steenrod algebra. We also need a Bockstein-free version of these algebra*
*s which we
denote by C0r.
If any of the above algebras with trivial Bockstein appears as the mod p coh*
*omology of
some space X it makes sense to ask about the order of the higher Bockstein whic*
*h connects
the polynomial and the exterior part and we indicate this order as a superscrip*
*t. In this
way, the notation
H*(X; Fp) ~=A(k)r
means that H*(X; Fp) ~=A0ras algebras over the Steenrod algebra and
ae0; i k
fi(i)(x) =
y; i = k + 1:
where fi(i)denotes the i-th order Bockstein, i.e. the i-th differential in the *
*mod p Bockstein
spectral sequence of X. In the same way, we introduce the notations H*(X; Fp) *
*= B(k)i;r
and H*(X; Fp) = C(k)r.
For further reference, we summarize the algebras that we have considered so *
*far in table
10.1.
HOMOTOPY CLASSIFICATION 9
3. Computing Lannes T functor. Let T denote the Lannes functor defined as left
adjoint to H - in the category U of unstable modules over the Steenrod algebra *
*(see [17]
for a full description of its properties.) Here H denotes the mod p cohomology *
*of Z=p as
in the previous section. When R is an unstable algebra over the Steenrod algebr*
*a then so
is T (R) and T becomes a functor in the category K of unstable algebras over th*
*e Steenrod
algebra.
Given a K-map f: R ! H, its adjoint restricts to a K-map T 0(R) ! Fp, where *
*T 0(R) is
the subalgebra of T (R) of all elements of degree zero. We define the connected*
* component
of T (R) corresponding to f as:
Tf(R) = T (R) T0(R)Fp:
Furthermore, Tf may be thought as a functor defined on the category of R-U-modu*
*les and
with values in the category of Tf(R)-U-modules (cf. [13].) We can also consider*
* Tf(M)
as an R-U-module induced by the natural K-map ": R ! Tf(R) and then ": M ! Tf(M)
becomes a natural transformation of R-U-modules.
The purpose of this section is to compute for various P E-algebras A constru*
*cted in the
previous section, namely, Ar for r|(p - 1) and Bi;rfor i 0 and r|(p - 1), the *
*particular
component of T (A) that corresponds to a map f: A ! H that can be uniformly des*
*cribed
as the composition
h k
A!- Fp[x]!- H
where h: A ! Fp[x] is the projection onto the polynomial part of A and k is the*
* obvious
inclusion of Fp[x] in the even part of H. Our result is as follows.
Theorem 3.1. (1) Tf(A1) ~=C1 and the natural map ": A1 ! Tf(A1) is the inclusi*
*on
of algebras given by "(x) = x and "(y) = xz.
(2) For all i 0 the natural map ": Bi;1-! Tf(Bi;1) is an isomorphism.
(3) For any r|(p - 1) and all i 0, the inclusions Ar ! A1 and Bi;r! Bi;1in*
*duce
isomorphisms Tf(Ar) ~=Tf(A1) ~=C1 and Tf(Bi;r) ~=Tf(Bi;1) ~=Bi;1.
We will be using the following lemma that can be easily obtained:
Lemma 3.2. Let A and B be two unstable algebras over the Steenrod algebra and f*
*: A !
B a K-map that induces an isomorphism Hom K(B; H) ~=Hom K (A; H): Then, for a K*
*-map
g: B ! H and any B-U-module M
TgOf(M) ~=Tg(M)
and the TgOf(A)-U-module structure of TgOf(M) is induced by TgOf(A) ! Tg(B). Th*
*ere-
fore, TgOf(M) is an A-U-module through A ! TgOf(A) ! Tg(B) or equivalently thro*
*ugh
A ! B ! Tg(B).
Proof of 3.1.(1). Recall that A1 = Fp[x] E[y], deg(x) = 2, fi(x) = y, P 1(y) =*
* yxp-1 and
f has been defined as the composition k O h where h is the projection A1!- Fp[*
*x] and k
identifies Fp[x] with the even part of H.
10 J. AGUADE, C. BROTO AND D. NOTBOHM
A1 sits in an exact sequence of A1-U-modules
(1) 0 ! yFp[x] ! A1 ! Fp[x] ! 0
and Tf is exact so our first job will be the calculation of Tf(yFp[x]) and Tf(F*
*p[x]). Both
yFp[x] and Fp[x] can actually be considered as Fp[x]-U-modules, with the A1-U-m*
*odule
structure induced by the projection h: A1 ! Fp[x]. According to lemma 3.2 what *
*we have
to do is to compute Tk(yFp[x]) and Tk(Fp[x]) as Fp[x]-U-modules. T (Fp[x]) is w*
*ell known
(see [3]) and ": Fp[x] ! Tk(Fp[x]) turns out to be an isomorphism. For yFp[x] w*
*e obtain:
Lemma 3.3. Tk(yFp[x]) ~=zFp[x] with deg(z) = 1; that is, a Fp[x]-U-module on on*
*e gen-
erator of degree one on which the Steenrod operations act trivially. Moreover, *
*": yFp[x] !
Tk(Fp[x]) is an Fp[x]-U-module map given by "(y) = zx.
Proof. yFp[x] might be identified to xFp[x] as Fp[x]-U-module ( denotes the sus*
*pension).
Since Tk commutes with suspensions we must calculate Tk(xFp[x]) and for this we*
* use the
following exact sequence of Fp[x]-U-modules:
0 ! xFp[x] ! Fp[x] ! Fp! 0:
Tk(Fp) is clearly trivial and we obtain Tk(xFp[x]) ~=Tk(Fp[x]) ~=Fp[x]. It also*
* follows that
" is the inclusion xFp[x] ! Fp[x].
Finally we apply and write Fp[x] as zFp[x] in order to get to the conclusio*
*n of the
lemma.
The above computation together with lemma 3.2 give us Tf(yFp[x]) and Tf(Fp[x*
*]) as
Tf(A1)-U-modules and also as A1-U-modules. Then, the exact sequence (1) induce*
*s a
diagram of A1-U-modules:
0 ----! yFp[x]----! A1 ----! Fp[x]----! 0
? ? ?
"?y "?y "?y~=
0 ----! zFp[x]----! Tf(A1) ----! Fp[x]----! 0
where the bottom row is an exact sequence of Tf(A1)-U-modules. This diagram imp*
*lies
that ": A1 ! Tf(A1) is a K-monomorphism and Tf(A1) ~=Fp[x] E[z] with " determi*
*ned
by "(x) = x and "(y) = xz.
Proof of 3.1.(2). Now we deal with the cases Bi;1~=Fp[xi] E[yi], degxi= 2pi, f*
*i(xi) = yi
and P pi(yi) = 0, so that Bi;1sits in an exact sequence of Bi;1-U-modules:
(2) 0 ! yFp[xi]!- Bi;1-! Fp[xi] ! 0
with yFp[xi] isomorphic as Fp[xi]-U-module to 2pi+1Fp[xi]. In these cases f = k*
* O h with h
the projection Bi;1-! Fp[xi] and k: Fp[xi] ! H defined by k(xi) = vpi, v a two *
*dimensional
generator of H.
HOMOTOPY CLASSIFICATION 11
Just as in the proof of the first part it is enough to compute Tk(yFp[xi]) a*
*nd Tk(Fp[xi])
and, in this case, both ": yFp[xi]!- Tk(yFp[xi]) and ": Fp[xi]!- Tk(Fp[xi]) a*
*re isomorphisms,
thus the sequence (2) gives rise to the diagram:
0 ----! yFp[xi]----! Bi;1 ----! Fp[xi]----! 0
? ? ?
"?y~= "?y "?y~=
0 ----! yFp[xi]----! Tf(Bi;1) ----! Fp[xi]----! 0
and this implies the desired result.
Proof of 3.1.(3). We will work out only the case of Ar for the case of Bi;ris e*
*ssentially
the same. Recall that Ar is the subalgebra of invariants of A1 by the action of*
* Z=r and T
commutes with taking invariants. In fact, we obtain:
Z=r Y Z=r
(1) Tf(Ar) ~=Tf A1 ~= Tf A1
f
where f runs through maps A1 ! H that restrict as f to Ar; that is, f is the *
*composition
k
A1 ! Fp[x] -! H, with k (x) = v, v a two dimensional generator of H and 2 Fp,*
* such
that f |Ar = f, and this equality holds if and only if r = 1, i.e. 2QZ=r Fp*.*
* Now it is
clear that an element of Z=r induces a permutation of the factors in 2Z=r Tf *
*(A1) and
then Y
Tf (A1) Z=r~= Tf(A1):
2Z=r
In a forthcoming section we will need a few variants of theorem 3.1.
Proposition 3.4. Let c : Bi;r! H, c : Ar ! H denote the homomorphisms which are
zero in positive degrees. Then Tc(Bi;r) ~=Bi;rand Tc(Ar) ~=Ar.
Proof. The proof is completely analogous to the proof of 3.1 except for two dif*
*ferences: We
have Tc(Fp) = Fpand this implies Tc(A1) = A1 and Tc(Bi;1) = Bi;1by the same arg*
*ument
as in 3.1(1) and (2). On the other hand, the product in (1) has only one factor*
* in the case
of the trivial homomorphism c.
One can check that the proofs of 3.1 and 3.4 work also for the algebras A0ra*
*nd B0i;r
and we have:
Proposition 3.5. Let c and f be as in 3.1 and 3.4 respectively. Then Tc(B0i;r) *
*~=B0i;r,
Tc(A0r) ~=A0rand Tf(A0r) ~=C01.
4. Non-realizability of P E-algebras. In this section we prove the Cooke conjec*
*ture as
stated in the introduction. The proof will follow from a study of the transgres*
*sion in the
Serre spectral sequence of some fibration. We start with a lemma describing an *
*interesting
relation in the action of the Steenrod algebra on the mod p cohomology of B2Z=p*
*. We use
the notation j j-1
P j = P pP p . .P.1:
12 J. AGUADE, C. BROTO AND D. NOTBOHM
Lemma 4.1. The following identities hold in H*(B2Z=p; Fp):
(1) P tP r fi = 0 for 0 < t < pr+1.
(2) P pjfiP j-1 fi = fiP j fi 6= 0 for j > 0.
Proof. Recall that H*(B2Z=p; Fp) is a free graded-commutative algebra on free g*
*enerators
P I where I is an admissible sequence of excess 2:
H*(B2Z=p; Fp) = Fp[; fiP 1fi; : :;:fiP j fi; : :]: E(fi; P 1fi; : :;:P j f*
*i; : :)::
Hence fiP j fi is an indecomposable in H*(B2Z=p; Fp). We prove first (1). If *
*r = 0 we
have P tP 1fi = P t+1fi = 0 by unstability. We proceed then by induction:
r [t=p]Xpr+t-s s
P tP r fi = P tP pP r-1 fi = sP P P r-1 fi:
s=0
In the right hand expression the term for s = 0 vanishes by unstability and all*
* other terms
arejzerojby-the1induction hypothesis. The lemma follows now from the Adem relat*
*ion for
P pfiP p in the following way
j-1
j pj-1 pX pj+pj-1-t t
P pfiP P j-2 fi= tfiP P P j-2 fi
t=0
pj-1-1X j j-1
+ tP p +p -tfiP tP j-2 fi:
t=0
(If j = 1, delete P j-2 in this formula.) By (1) and the unstability condition,*
* the right
hand term reduces to pj-1fiP pjP j-1 fi and the proof ends by checking pj-1= 1.
Proposition 4.2. Assume H*(X; Fp) ~= Bi+1;1, i 0, as algebras over the Steenrod
algebra. Assume also that there is a fibration
g
X!- E!- B2Z=p
such that x transgresses to P i fi plus decomposables, where 2 H2(B2Z=p; Fp) i*
*s the
fundamental class. Then i = 0.
Proof. Consider the spectral sequence of the fibration X ! E ! B2Z=p. Since o(*
*x) =
P i fi + d this element has to be killed by g*. Hence g*(fiP i fi + fid) = 0. I*
*f we assume
i > 0 we can apply the lemma and obtain
i 1 * pi+1 1 * p 1
0 = g*(P 1P pfiP i-1fi+P fid) = g (P fiP i-1fi+P fid) = g ([fiP i-1fi] +P fi*
*d):
Notice that fiP i-1fi is an even dimensional indecomposable in H*(B2Z=p; Fp). U*
*sing 4.1
we have the equalities
P 1P j fi= 0; j 0;
ae[fiP j-1 fi]p; j > 0
P 1fiP j fi=
0; j = 0
HOMOTOPY CLASSIFICATION 13
and an elementary argument shows that P 1fid cannot contain the term [fiP i-1fi*
*]p.
Hence, there is some differential in the spectral sequence, coming from an elem*
*ent in
total degree 2pi+1+ 2p - 1 which kills [fiP i-1fi]p + other terms. By inspectin*
*g the E2-
term of the spectral sequence in total degree 2pi+1 + 2p - 1 we see that only x*
* p-2fi
and y p-1 may eventually kill this element.
We know by hypothesis that the first non vanishing differential maps x to P *
*i fi + d.
Hence it does not map x p-2fi to [fiP i-1fi]p + other terms. Since fix = y, th*
*e next
differential maps y to fiP i fi + fi(d) and so it cannot kill [fiP i-1fi]p + ot*
*her terms. In
any case, [fiP i-1fi]p + other termssurvives, a contradiction that can only be *
*avoided if
i = 0.
Notice that the "elementary argument" mentioned in the above proof fails if *
*p = 2.
This fact gives rise to a manifold of fascinating phenomena which will be studi*
*ed in [4].
Theorem 4.3. If H*(X; Fp) ~=Bi;ras algebras over the Steenrod algebra, then i *
*1.
Proof. Since H1(X; Fp) = 0 we can assume, without loss of generality, that X is*
* p-complete.
Let f : Bi;r! H*(BZ=p; Fp) be the non trivial homomorphism considered in the la*
*st
section. Then by [17; 3.1.1] there is a map OE : BZ=p ! X inducing f in mod p c*
*ohomology.
By theorem 3.1 we have
TfBi;r~=Bi;1;
where T denotes the T functor with respect to V = Z=p. Then, [17; 3.2.1] shows *
*that
H*(map (BZ=p; X)OE; Fp) ~=Bi;1
where map (BZ=p; X)OEis the space of all maps BZ=p ! X homotopic to OE. Observe*
* now
that BZ=p is a connected abelian simplicial group and the action of BZ=p on its*
*elf by
right translations induces an action of BZ=p on the space map (BZ=p; X)OE. If *
*Y is the
homotopy quotient of map (BZ=p; X)OEby this action, we have a fibration
(1) map (BZ=p; X)OE! Y ! B2Z=p:
If we denote by i the induced map BZ=p ! map (BZ=p; X)OEand by e : map (BZ=p; X*
*)OE!
X the evaluation map at the base point of BZ=p (which is the unit of BZ=p as a *
*simplicial
group) then one sees easily that e O i = OE. In particular, i*(x) = vpi = P i-*
*1fiu where
x 2 Bi;1is the class in degree 2pi. Hence, the class x in the mod p cohomology*
* of the
fibre of (1) transgresses to P i-1fi plus decomposables and proposition 4.2 sho*
*ws that
i 1.
Remark 4.4. This method can be applied to give a new short proof of the well kn*
*own fact
that if H*(X; Fp) ~=Fp[x2pi] then i = 0 for p odd and i = 0; 1 for p = 2. In th*
*is case there
exists also a fibration sequence
__ g 2
BZ=p!- X!- X!- B Z=p:
Then o(x) = P i fi2 + d where d is a decomposable and the result follows from 0*
* =
g*(fiP i fi2 + fid).
14 J. AGUADE, C. BROTO AND D. NOTBOHM
5. Spaces realizing Ar. Let us denote by ss the additive group of the p-adic in*
*tegers.
Let G be the automorphism group of ss. G is isomorphic to the multiplicative gr*
*oup of the
invertible elements in the ring structure of ss and this is the direct product *
*of the cyclic
group of (p - 1)th-roots of unity by the group U1 = 1 + pss. The group U1 is to*
*rsion free
and there is a monomorphism
OE : ss ! G
given by OE(x)(y) = exp(px)y where the product is taken in the ring structure o*
*f the p-adic
integers. Moreover, after identifying G with the invertibles of ss, OE maps on*
*to U1, the
logarithm providing an inverse.
We obtain therefore a precise description of all possible actions of ss on s*
*s, namely, all
these actions are obtained by composing OE with multiplication by a p-adic numb*
*er ff. We
will denote by OEff the one defined by ff and by ssOEffthe additive group of p-*
*adic integers
endowed with the action defined by OEff. Among them the ones of most interest *
*for us
correspond to ff = pk for k 0 and we will abbreviate OEpk as OEk.
Realizing Ar. We will construct a space realizing Ar as well as other related s*
*paces. We
suggest to consider tables 10.2 and 10.3 in the appendix as a quick reference g*
*uide to the
spaces introduced in this section. For this aim we consider ssOEk, the p-adics *
*endowed with
the action defined by OEk for k 0. B2ssOEkinherits the action and we define sp*
*aces
Ek = EOEk= B2ssOEkxssEG
for all k 0. (ss acts on G through OE.) Let us compute the mod p cohomology *
*of Ek.
From the obvious fibration
B2ssOEk! Ek ! EG=ss ' Bss
we get a spectral sequence
H*(ss; H*(B2ssOEk; Fp)) ) H*(Ek; Fp):
Notice that B2ss ' BS1bpand Bss ' S1bp. Since ss is q-divisible for any q 6= p,*
* ss can only
act trivially on H*(B2ssOEk; Fp) which is either trivial or one-dimensional in *
*each degree.
Hence the spectral sequence yields immediately that for any k 0
H*(Ek; Fp) ~=E(z) Fp[x]
where z and x are classes in degrees 1 and 2, respectively. The Steenrod algebr*
*a should
act trivially on z and the Steenrod powers act on Fp[x] as they do in H*(BS1; F*
*p). It only
remains to determine the action of the Bockstein homomorphism on the class x. T*
*his will
distinguish E0 from Ek for k 1. More in general, we will show that the action*
* of the
higher Bocksteins on x implies that all these spaces are different.
Proposition 5.1. H*(Ek; Fp) ~=C(k)1for k 0.
Proof. To prove this, we will compute the cohomology of Ek with p-adic coeffici*
*ents in low
dimensions by means of the Serre spectral sequence. We need the following resul*
*ts on the
homology of the p-adic integers.
HOMOTOPY CLASSIFICATION 15
Lemma 5.2. (1) H1(ss; Z) = ss and for j 2, Hj(ss; Z) is a Q-vector space.
(2) H1(ss; ss) ~=ss and for j 2, Hj(ss; ss) = 0 (trivial coefficients).
(3) The cohomology of ss with twisted p-adics coefficients is
H0(ss; ssOEff)= 0;
H1(ss; ssOEff)~=Z=p(ff)+1;
Hj(ss; ssOEff)= 0; j 2
where (ff) denotes the biggest power of p dividing ff.
Proof. We obtain H1(ss; Z) ~=ss by the Hurewicz theorem. Since Hj(ss; Z=q) = 0*
* for all
j 2 and all primes q the universal coefficient formula implies that Hom (Hj(ss*
*; Z); Z=q) ~=
Ext (Hj(ss; Z); Z=q) ~=0 for all j 2 and all primes q, hence statement (1) fol*
*lows.
The statement (2) follows by the universal coefficient formula because Hom (*
*ss; ss) ~=ss,
Hom (A; ss) = 0 if A is p-divisible and Ext(A; ss) = 0 if A is torsion free.
To prove (3) note first that zero is the only invariant element of ssOEffund*
*er the action of
ss so H0(ss; ssOEff) = 0. Next, we consider the well known description of the f*
*irst cohomology
group through derivations:
H1(ss; ssOEff) ~=Der(ss; ssOEff)=Ider(ss; ssOEff):
A derivation ss ! ssOEffis determined by the image of 1 2 ss. In fact, for a gi*
*ven derivation
d: ss ! ssOEff, if x 2 ss then d(x) + exp(pffx)d(1) = d(x + 1) = d(1 + x) = d(1*
*) + exp(pff)d(x)
and this equation has a unique solution for d(x) once d(1) is fixed.
Moreover, the formula
da(x) = a exp(pffx)_-_1
p(ff)+1
defines a derivation ss ! ssOEfffor any a 2 ss. This derivation is inner preci*
*sely when
a 0 (p(ff)+1) and therefore H1(ss; ssOEff) ~=Z=p(ff)+1.
It remains to compute Hj(ss; ssOEff) for j 2. We will see that H*(ss; ssOEf*
*f) is isomorphic
to H*(Z; ssOEff) with the action induced by restriction and then the result wil*
*l follow because
Z is free.
The isomorphism that we claim is induced by the inclusion Z ! ss and it is p*
*roved in
degrees 0 and 1 by direct computation. It would be also clear if the coefficien*
*ts were Z=pr
for any r > 1. Then the Lyndon-Hochschild-Serre spectral sequence for Z ! ss !*
* ss=Z
shows first that "H*(ss=Z; Z=pr) = 0 and since ss=Z can only act trivially on Z*
*=pr, also that
H*(ss; ssOEff) ~=H*(Z; ssOEff).
As a consequence of this lemma, in the spectral sequence of the fibration B2*
*ssOEk!
Ek ! Bss with coefficients in ^Zp the only term that can contribute to H3(Ek; ^*
*Zp) is
H1(ss; H2(B2ss; ^Zp)) ~=Z=pk+1. This finishes the proof of proposition 5.1.
Notice now that for any r dividing p - 1 there is an embedding of Z=r in G w*
*hich gives
an action of Z=r on B2ss and EG. Since G is isomorphic to Z=p - 1 x ss we obta*
*in an
induced free action of Z=r on Ek. Notice that since fi(k+1)(x) = xz the action *
*has to be
16 J. AGUADE, C. BROTO AND D. NOTBOHM
trivial on z 2 H1(Ek; Fp). Let Ek(r) be the quotient of Ek by this action. Sinc*
*e r is prime
to p, it is clear that
H*(Ek(r); Fp) = H*(Ek; Fp)Z=r = E(z) Fp[u]
where u corresponds to xr in H*(Ek; Fp) and fi(k+1)(u) = ruz. Hence,
Proposition 5.3. H*(Ek(r); Fp) ~=C(k)rfor k 0, r|(p - 1).
Finally, let us consider the composition
f
Bss!- Ek!- Ek(r)
where f is a section of the fibration B2ss ! Ek ! Bss. If E0k(r) denotes the co*
*fibre of this
composition then we have H*(E0k(r); Fp) ~=Fp[u] E(w) with degu = 2r, degw = 2r*
* + 1,
fi(k+1)(u) = w and P 1(w) = rusw, where s = (p - 1)=r as is usual in this paper*
*. In
particular
H*(E0k(r); Fp) ~=A(k)r:
Definition 5.4. For k 0, r|p - 1, we define Xk(r) as the p-completion of E0k(r*
*).
The next theorem establishes some properties of these spaces.
Theorem 5.5. For any r dividing p - 1 and k 0,
(1) Xk(r) is a simply connected p-complete space whose homotopy groups are *
*finite
p-groups.
(2) H*(Xk(r); Fp) ~=A(k)r.
Proof. Will be based in the following two propositions.
Proposition 5.6. Let R = ^Zpor Z and X a space with cohomology of finite type o*
*ver R.
If H*(X; Fp) = A(k)rthen in the R-cohomology Bockstein spectral sequence {Bl; d*
*l} for X
(1) thenfirst non-trivial differential is dk+1nand dk+1(u)n= w,
(2) up survives to Bn+k+1 and dn+k+1 ([up ]) = [up -1w] and
(3) B1 = 0.
Proof. This is a direct consequence of known results about the differential in *
*the Bockstein
spectral sequence (cf. [16; pag. 102]).
The next proposition might be of independent interest and we establish it fo*
*r any prime
number, either two or odd.
Proposition 5.7. Let p be any prime and X a 1-connected, p-complete space, then:
(1) The following conditions are equivalent:
(i) Hj(X; Fp) is finite for all j.
(ii)ssj(X) is a finitely generated ^Zp-module for all j.
(iii)Hj(X; ^Zp) is a finitely generated ^Zp-module for all j.
(2) The following conditions are equivalent:
(i) "Hj(X; ^Zp) is a finite p-group for all j.
(ii)ssj(X) is a finite p-group for all j.
HOMOTOPY CLASSIFICATION 17
Proof. Let F be the fibre of the rationalization X ! X0. We can first obtain so*
*me general
facts about F . H*(F ; Fp) ~= H*(X; Fp) and F ! X is the p-completion of F . *
* Also,
H*(F ; ^Zp) ~=H*(X; ^Zp) and ssj(F ) is a p-group for all j. Moreover, since F *
*is the fibre
of a map between 1-connected spaces, the fundamental group of F is abelian and *
*acts
trivially on the homology and homotopy of the universal cover of F (cf. [15]). *
*Hence the
mod C Hurewicz theorem can be applied to F and we obtain that H"j(F ; Z) is a p*
*-group
(i.e. a group all of whose elements are p-power torsion) for any j.
Now, the proof of part (1) of the proposition will consist in the following *
*sequence of
statements.
Claim 5.7.1: Hj(X; Fp) is finite for all j if and only if H"j(F ; Z) is a finit*
*ely cogenerated
p-group for all j.
Let us write H"j(F ; Z) as an extension of a divisible p-group Dj by a pure *
*subgroup
Pj which is a direct sum of cyclic p-groups. Then one easily deduces that the *
*mod p
homology of F is of finite type over Fpif and only if both Pj and Dj contain fi*
*nitely many
summands for all j. Since a bounded pure subgroup is a direct summand this mea*
*ns
that the (reduced) integral homology groups of F are a direct sum of finitely m*
*any cyclic
p-groups and finitely many groups Z=p1 ; that is, they are finitely cogenerated*
* p-groups.
Claim 5.7.2: H"j(F ; Z) is a finitely cogenerated p-group for all j if and only*
* if ssj(F ) is so.
Since ss1(F ) is abelian ss1(F ) ~=H1(F ; Z) and then since the class of fin*
*itely cogenerated
abelian p-groups is an acyclic ring of abelian groups this claim follows by the*
* mod C
Hurewicz theorem.
Claim 5.7.3: ssj(F ) is a finitely cogenerated p-group for all j if and only i*
*f ssj(X) is a
finitely generated ^Zp-module for all j.
From the homotopy exact sequence for the fibration F ! X ! X0 we obtain short
exact sequences
0 ! ssj+1(X) Q=Z ! ssj(F ) ! Tor(ssj(X); Q=Z) ! 0
and then one of the implications. On the other hand, since X is the p-completio*
*n of the
nilpotent space F we also have short split exact sequences ([6; VI.5.1])
0 ! Ext(Z=p1 ; ssj(F )) ! ssj(X) ! Hom (Z=p1 ; ssj-1(F )) ! 0;
hence the implication in the other direction is also true.
Claim 5.7.4: Hj(X; Fp) is finite for all j if and only if Hj(X; ^Zp) is a finit*
*ely generated
^Zp-module for all j.
It suffices to show this equivalence for F , and this follows easily by the *
*universal coeffi-
cients formula using claim 5.7.1.
This finishes the proof of part (1). Let us turn to the proof of part (2). W*
*ith the same
argument as in Claim 5.7.4 we obtain that H"j(X; ^Zp) is a finite p-group for a*
*ll j if and
only if "Hj(F ; Z) is a finite p-group for all j. Again by the mod C Hurewicz t*
*heorem this is
equivalent to ssj(F ) to be a finite p-group for all j and finally the same arg*
*ument of Claim
5.7.3 shows that if the homotopy groups of either X or F are finite p-groups, t*
*hen F ! X
is actually a homotopy equivalence.
18 J. AGUADE, C. BROTO AND D. NOTBOHM
We can now finish the proof of Theorem 5.5. The space Xk(r) was defined as *
*the p-
completion of E0k(r). By construction we have ss1(Ek(r)) ~=Z=r x ss and then by*
* the van
Kampen theorem ss1(E0k(r)) ~=Z=r. Hence ([6; p. 206]) E0k(r) is Z=p-good and Xk*
*(r) is
p-complete, simply connected and has the same mod p cohomology as E0k(r). So we*
* have
proved part (2) of the theorem.
By Proposition 5.7(1) Xk(r) is of finite type over ^Zp hence the Bockstein s*
*pectral
sequence applies and by Proposition 5.6 the cohomology groups "Hj(Xk(r); ^Zp) a*
*re actually
finite p-groups, hence by proposition 5.7(2) we obtain part (1) of the theorem.
Remark 5.8. From Proposition 5.6 we can derive the integral cohomology of the s*
*paces
Xk(r):
ae k+1+(j)
"Hi(Xk(r); ^Zp) ~=H"i(Xk(r); Z) ~= Z=p i = 2rj + 1; j 1
0 otherwise.
Final remarks. Here is the reason for which we have been dealing with a certain*
* collection
among all possible actions of ss on ss.
Proposition 5.9. Let ss be the additive group of the p-adic integers together *
*with a ss
action defined by a non trivial homomorphism : ss ! Aut(ss) = G and define
E = B2ss xssEG:
Then E is homotopy equivalent to a space Ek = EOEk
Proof. From our discussion of the possible actions of ss on ss at the beginning*
* of this section,
is of the form OEff for a p-adic integer ff. That is:
(x)(y) = epffxy:
Now, ff might be written as ff = p(ff)w where w 2 1 + pss. Since w is invertib*
*le it
determines an automorphism
w: ss ! ss:
Now, the identity B2ss ! B2ss is w-equivariant if we consider the action given *
*by on the
source an by OE(ff)on the target:
(ff)x
(x)(y) = epffxy = epwp y = OEk(wx)(y):
In this way we get a map E ! E(ff)which is in fact a homotopy equivalence beca*
*use w
is invertible.
Remark 5.10. Observe that until now all our constructions could be performed us*
*ing BZ
instead of Bss as base space of our fibrations, with actions of Z on ss induced*
* by restriction
from the actions of ss on ss that we used. Also, we could use BZ=p1 instead o*
*f B2ss.
However the above proposition would not be true in that case. We would need to *
*complete
our spaces before proving such a result.
HOMOTOPY CLASSIFICATION 19
6. Uniqueness of spaces realizing Ar. In section 5 we have constructed the spac*
*es
Xk(r) whose mod p cohomologies realize A(k)rfor r dividing p - 1 and k 0 (theo*
*rem
5.5). In this section we show that up to p-adic completion these spaces are the*
* only ones
which realize the algebras with higher Bocksteins A(k)r.
Theorem 6.1. Let X be such that H*(X; Fp) ~=A(k)rfor some r|(p - 1) and k 0. T*
*hen
X^p' Xk(r).
Proof. Let X be a space satisfying the hypothesis of the theorem. Since H1(X; F*
*p) = 0 we
have that ss1(X) is Z=p-perfect. Hence ([6; p. 206]) X^pis a simply connected p*
*-complete
space with the same mod p cohomology as X itself. By 5.6 and 5.7 the homotopy g*
*roups
of X^pare finite p-groups.
Let f : BZ=p ! X^p be a map such that f* is non-trivial in degree 2r and tri*
*vial
in degree 2r + 1 and let Y denote the component of the mapping space map (BZ=p;*
* ^Xp)
containing the map f. There is an evaluation map e : Y ! X^p. The next step *
*in the
proof of 6.1 will be to show that Y is homotopy equivalent to the space Ek of s*
*ection 5.
According to the computation of the T functor in section 3, Tf(H*(X; Fp)) ~=E(z*
*) Fp[w]
with deg(z) = 1, deg(w) = 2 and
aezw; k = 0
fi(w) =
0; k > 0:
Notice that Tf(H*(X; Fp)) is only an algebra over the Steenrod algebra and so h*
*igher Bock-
steins do not make any sense in Tf(H*(X; Fp)) unless we show that it is the coh*
*omology
of some space.
The computed value of the functor Tf is interpreted by [10] as follows. Let *
*PnX^p denote
the n-th stage of the Postnikov decomposition of X^p. Then map(BZ=p; PnX^p)fn*
* is a
tower with
Y ' lim-map(BZ=p; PnX^p)fn
n
and ae
C01; k > 0;
lim-!H*(map (BZ=p; PnX^p)fn; Fp) ~=
n C1; k = 0:
The natural homomorphism H*(X; Fp) ! lim-!nH*(map (BZ=p; PnX^p)fn; Fp) sends x *
*to wr
and y to wrz.
Some information about the homotopy of the spaces map (BZ=p; PnX^p)fn is pro*
*vided
by results of Thom ([26], revisited in [21]). The principal fibration PnX^p ! P*
*n-1X^p gives
rise to a principal fibration
map (BZ=p; PnX^p)fn ! map (BZ=p; Pn-1X^p)fn-1
with fibre a union of components of map (BZ=p; K(ssnX^p; n)). But each of these*
* compo-
nents has the homotopy type of a product of Eilenberg-MacLane spaces
K(Hn-j (BZ=p; ssnX^p); j); 1 j n:
20 J. AGUADE, C. BROTO AND D. NOTBOHM
Since the homotopy groups of X^pare finite p-groups and so are the homotopy gro*
*ups of
map (BZ=p; Pn-1X^p)fn-1 by induction, those of map (BZ=p; PnX^p)fn should also *
*be finite
p-groups.
Let us write Yn = map (BZ=p; PnX^p)fn.
The class z in degree one in lim-!nH*(Yn; Fp) is represented by a class zn 2*
* H*(Yn; Fp)
for some n and also by its images in H*(Yn+i; Fp). We fix such a sequence {zn}*
*. This
sequence provides a map of towers:
{zn}
{Yn} ---! {K(Z=pff(n); 1)}:
We will prove that we can choose the sequence ff(n) to be unbounded. In fact, *
*any of
these maps is a lifting of the classifying map Yn!- K(Z=p; 1) of the class zn.*
* Suppose
by induction that zn is classified by zn: Yn!- K(Z=pff(n); 1), that is z*n() =*
* zn if is the
fundamental class of H*(K(Z=pff(n); 1); Fp). Observe that we can as well assume*
* that ff(n)
is the maximum possible such that this lifting exists. This is because all of t*
*he homotopy
groups of Yn are finite and then zn should be dual to a torsion homology class.*
* Now we
look at the class zn+i, i 1. This is classified by Yn+i!- Yn!- K(Z=pff(n); 1*
*) and the
obstructions for the existence of a lifting
Yn+i!- K(Z=pff(n+i); 1)
with ff(n + i) > ff(n) are some higher Bocksteins. But no higher Bockstein can*
* be non
trivial on zn for all big enough n because if this happens then wr is in the im*
*age of some
higher Bockstein, contradicting the fact that x 2 H*(X; Fp) has a non trivial B*
*ockstein of
order k + 1. Hence, limn!1 ff(n) = 1.
Consider now the inverse system of fibrations
: :-:---! Fn ----! Fn-1 ----! : : :
?? ?
y ?y
: :-:---! Yn ----! Yn-1 ----! : : :
?? ?
y ?y
: :-:---! BZ=pff(n) ----! BZ=pff(n-1) ----! : : :
Notice that since the maps Yn!- BZ=pff(n)are liftings of non-trivial maps Y*
*n!- BZ=p,
they induce epimorphisms between fundamental groups and so the spaces Fn are co*
*nnected.
All homotopy groups involved in the above inverse system of fibrations are f*
*inite p-
groups, hence, in the limit, we get a fibration:
F!- Y!- Bss
where F = lim-nFn and ss denotes as usual the additive group of the p-adic inte*
*gers. Note
that in all these fibrations the base space is not simply connected. Neverthel*
*ess, at any
stage the Eilenberg-Moore spectral sequence of [12] starts with
*
E*;*2~=Tor*;*H*(BZ=pff(n);Fp)H (Yn; Fp); Fp
HOMOTOPY CLASSIFICATION 21
and converges strongly to the mod p cohomology of the fibre Fn because ([12]) t*
*he funda-
mental group of the base is a p-group and thus it acts nilpotently on the mod p*
* cohomology
of the fibre. In the limit we have a spectral sequence
*
E*;*2~=lim-!Tor*;*H*(BZ=pff(n);Fp)H (Yn; Fp); Fp
n
converging to lim-!nH*(Fn; Fp).
Lemma 6.3. lim-!nTor*;*H*(BZ=pff(n);Fp)H*(Yn; Fp); Fp ~=Fp[w], deg(w) = 2 and s*
*o therefore
lim-!nH*(Fn; Fp) ~=Fp[w].
Proof. Tor is covariant with respect to any of its three variables and lim-!is *
*an exact functor.
Hence one can easily derive a commutation formula for Tor and lim-!which shows *
*that
* *;*
lim-!Tor*;*H*(BZ=pff(n);Fp)H (Yn; Fp); Fp ~=TorE(z) E(z) Fp[w]; Fp ~=Fp*
*[w]:
n
Alternatively, one can directly compute the lim-!as follows.
Let us denote
* *
Kn = ker H (Yn; Fp) ! lim-!H (Yn; Fp) ~=E(z) Fp[w] :
n
Kn is an ideal of H*(Yn; Fp) and therefore a sub-H*(BZ=pff(n); Fp)-module. The *
*induced
H*(BZ=pff(n); Fp)-module structure of E(z) Fp[w] factors:
H*(BZ=pff(n); Fp)----! E(zn)
?? ?
y ?y
Kn ----! H*(Yn; Fp) ----! E(z) Fp[w]
Observe that H*(BZ=pff(n); Fp) ~=E(zn) Fp[an], deg(an) = 2. As a consequen*
*ce we
have:
(1) Tor*;*H*(BZ=pff(n);Fp)E(z) Fp[w]; Fp
~=Tor*;*E(zE(z ) F[w]; F Tor*;* (F ; F)
n) n p p Fp[an]p p
~=Fp[w] TorFp[an](Fp; *
*Fp)
There is an exact sequence:
r;s *
(2) : :!:Torr;sH*(BZ=pff(n);Fp)Kn; Fp ! TorH*(BZ=pff(n);Fp)H (Yn; Fp); Fp !
r-1;s
! Torr;sH*(BZ=pff(n);Fp)E(z) Fp[w]; Fp ! TorH*(BZ=pff(n);Fp)Kn; Fp*
* ! : : :
22 J. AGUADE, C. BROTO AND D. NOTBOHM
which is natural with respect to maps Yn+i ! Yn. Observe that no element of Kn *
*survives
to the limit lim-!H*(Yn; Fp). Since Kn is finite dimensional there is a large e*
*nough i such
that Kn ! Kn+i is zero, hence so is
*;*
Tor*;*H*(BZ=pff(n);Fp)Kn; Fp ! TorH*(Z=pff(n+i);Fp)Kn+i; Fp:
Then (2) implies
* *;*
lim-!Torr;sH*(BZ=pff(n);Fp)H (Yn; Fp); Fp ~=limTorH*(BZ=pff(n);Fp)E(z) Fp*
*[w]; Fp:
n -!n
For similar reasons (1) implies:
lim-!Tor*;*H*(BZ=pff(n);Fp)E(z) Fp[w]; Fp ~=Fp[w]:
n
Lemma 6.4. Let {Zn} be a tower of fibrations of pointed connected p-complete sp*
*aces
with mod p cohomology of finite type. If lim-!nH*(Zn; Fp) is a polynomial algeb*
*ra on one
generator w in degree 2 then lim-nZn ' B2ss.
Proof. An argument similar to one used above shows that there is a map of tower*
*s:
{gn}
(1) {Zn} ---! {K(Z=pfl(n); 2)}
with fl(n) an unbounded sequence. Here, each gn detects a class in degree two o*
*f H*(Zn; Fp)
that represents w in the limit, hence {gn} induces an isomorphism
lim-!H*(Zn; Fp) ~=limH*(K(Z=pfl(n); 2); Fp) ~=Fp[w]
n -!n
and dually
lim-H*(Zn; Fp) ~=limH*(K(Z=pfl(n); 2); Fp)
n -n
because all relevant (co)homology groups are finite. By the same reason, this i*
*mplies that
the induced map of towers:
{H*(Zn; Fp)} ! {H*(K(Z=pfl(n); 2); Fp)}
is a pro-isomorphism.
Now, according to [6; III.6.6, pg. 88] the map of towers {RnZn} ! {RnK(Z=pfl*
*(n); 2)} is
a weak pro-homotopy equivalence, where {RnX} is the tower which defines Bousfie*
*ld-Kan
p-completion. Hence,
lim-Zn = limR1 Zn = limRnZn ' limRnK(Z=pfl(n); 2) = B2ss:
n -n -n -n
HOMOTOPY CLASSIFICATION 23
This lemma applies immediately to the tower {Fn} because the homotopy groups*
* of
each Fn are finite p-groups and a space whose homotopy groups are finite p-grou*
*ps is
necessarily p-complete, nilpotent and of finite mod p type. We have, therefore,*
* obtained a
fibration
(2) B2ss ! Y ! Bss:
Since ss can only act trivially on H*(B2ss; Fp), Y is p-complete (cf. [6; mod*
*-Z=p fibre
lemma]). The Serre spectral sequence and the injectivity of H*(X; Fp) ! lim-!H**
*(Yn; Fp)
show that the natural map
H*(Y ; Fp) ! lim-!H*(Yn; Fp) ~=Tf(H*(X; Fp))
n
is an isomorphism. By naturality of the Bockstein homomorphisms, we deduce that
H*(Y ; Fp) = C(k)1. We want to deduce from here that the fibration (2) is fibr*
*e homo-
topy equivalent to the fibration
B2ss ! Ek ! Bss
of section 5. Fibrations with base space Bss and fibre B2ss are classified by t*
*he homotopy
set:
Bss; B Aut(B2ss)
where Aut (B2ss) is the topological monoid of the self homotopy equivalences of*
* B2ss.
According to [22] there is a fibration
B2ss ! B Aut(B2ss) ! B Aut(ss)
having a section B Aut(ss) ! B Aut(B2ss). Then
2
Bss; B Aut(B ss) ~= Bss; B Aut(ss) ~=Hom (ss; Aut(ss)):
Therefore, any fibration B2ss ! Z ! Bss is determined by an action of ss on *
*ss. All these
actions were considered in section 5. From such classification we obtain an equ*
*ivalence of
fibrations
B2ss ----! Y ----! Bss
flfl x x
fl '??g '??w
B2ss ----! Ek ----! Bss
There is an action of Z=r on Ek considered in section 5 and also an action o*
*f Z=r on
Y defined in the following way. We have Y = map (BZ=p; ^Xp)f and Z=r acts on *
*BZ=p.
Since f* commutes with this action, we get an action on Y such that the evaluat*
*ion map
e : Y ! ^Xpis equivariant. Naturality of T shows that on mod p cohomology this *
*action
leaves z fixed and sends w to w where is an r-th root of unity. In this form w*
*e obtain
a map l : Ek ! ^Xp
24 J. AGUADE, C. BROTO AND D. NOTBOHM
Lemma 6.5. l is homotopic to an equivariant map.
Proof. First of all we notice that it is enough to prove that l is equivariant *
*up to homotopy
for Wojtkowiak proved in [27] that when a finite group of order prime to p acts*
* freely on a
space and the target space is p-complete, nilpotent and of finite type over ^Zp*
*then a map
equivariant up to homotopy is homotopic to an equivariant map.
Ek is a two stage Postnikov system and there is an exact sequence of Didierj*
*an ([8]) for
the group of homotopy classes of self homotopy equivalences of Ek:
1 ! H2(ss; ss'k) ! E(Ek) ! Aut(ss) Aut(ss):
Since H2(ss; ss'k) = 0, this shows that a homotopy self equivalence of Ek is de*
*termined up
to homotopy by its action on ss1(Ek) and ss2(Ek). The lemma is proved if we sho*
*w that
Z=r acts on ss1(Y ) and ss2(Y ) as it does on ss1(Ek) and ss2(Ek).
The action of Z=r on ss1(Y ) = ss is determined by the action on H1(Y ; Fp) *
*which can
only be trivial. Similarly, the action of Z=r on ss2(Y ) = ss is determined by *
*the action on
H2(Y ; ^Zp) ~=^Zpand this action is determined by the action on H2(Y ; Fp).
Hence we obtain a map
h : Ek(r) = Ek xZ=rEZ=r ! ^Xp:
Let now k : Bss ! Ek(r) be the map considered in section 5. If kh is trivial we*
* obtain a map
E0k(r) ! ^Xpwhich induces isomorphism in mod p cohomology and the theorem is pr*
*oved.
But ^Xpis simply connected, its homotopy groups are finite p-groups and Hi(ss; *
*P ) = 0 for
i > 1 and any finitely generated ^Zp-module P with trivial action. Hence, by ob*
*struction
theory, any map Bss ! ^Xpis trivial. This ends the proof of theorem 6.1.
7. Spaces realizing Bi;r. In section 2 we constructed for each algebra Bi;ra to*
*pological
realization. It turns out that these are not the only ones. In this section we *
*will construct
several families of spaces, some of which will have cohomology isomorphic to Bi*
*;r. We
suggest using tables 10.2 and 10.3 in the appendix as a quick reference guide t*
*o all these
spaces.
Let Xk = Xk(1) be the spaces introduced in section 5. Theorem 5.5 proves that
H*(Xk; Fp) ~=A(k)1:
Hence, the two dimensional class x in H*(Xk; Fp) can be represented by a map
(1) Xk ! B2Z=pk+1:
Let Yk be the fibre of this map. Yk is a p-complete space with finite homotopy*
* groups.
From the construction of Xk we see that there is an action of the cyclic group *
*of order r on
Xk, for any r|p - 1. By [27] we can assume that the map (1) is equivariant with*
* respect to
this action and the natural action on B2Z=pk+1. This yields an action of the cy*
*clic group
of order r on Yk and we define Yk;ras the p-completion of the homotopy quotient*
* of Yk by
this action:
Yk;r= (EZ=r xZ=rYk)bp:
HOMOTOPY CLASSIFICATION 25
Proposition 7.1. H*(Yk;r; Fp) ~=B1;ras algebras over the Steenrod algebra.
Proof. We consider the sequence of fibrations
BZ=pk+1 ! Yk;1! Xk ! B2Z=pk+1:
In the spectral sequence of the first three terms, the classes u and v in H*(BZ*
*=pk+1; Fp) are
transgressive and are mapped onto the classes x and y of H*(Xk; Fp). By degree *
*reasons
it follows that these are the only non vanishing differentials. Therefore, H*(Y*
*k;1; Fp) is a
P E-algebra of type (2p; 2p + 1).
We prove now H2p+1(Yk;1; ^Zp) ~=Z=p.
We have seen that Yk;1, being a p-complete space, is (2p - 1)-connected. All*
* homology
groups of Yk;1are torsion groups, and therefore, H2p(Yk;1; ^Zp) = 0. The long *
*exact se-
quence of cohomology groups associated to the fibration Yk;1! Xk ! B2Z=pk+1 con*
*tains
0!- H2p+1(B2Z=pk+1; ^Zp)!- H2p+1(Xk; ^Zp)
!- H2p+1(Yk;1; ^Zp)!- H2p+2(B2Z=pk+1; ^Zp)!- H2p+2(Xk; ^Zp) = *
*0:
The last group vanishes because of remark 5.8. The first two groups are isomor*
*phic,
because both measure which high order Bockstein acts nontrivially on 2, the gen*
*erator
of H*(B2Z=pk+1; Z=p), or on xp. In both cases this is fi(k+2). Thus, we have *
*to cal-
culate H2p+2(B2Z=pk+1; ^Zp). In dimension 2p + 1, the mod-p cohomology of B2Z=*
*pk+1
is generated by p-12fi(k+1)(2) and P 1fi(k+1)(2). All higher order Bocksteins *
*vanish on
p-12fi(k+1)(2), which therefore comes from an integral class, and fiP 1fi(k+1)(*
*2) 6= 0. Thus
H2p+2(B2Z=pk+1; ^Zp) ~=Z=p and H2p+1(Yk;1; ^Zp) ~=Z=p as claimed.
Hence, the two generators of H*(Yk;1; Fp) are connected via the Bockstein. *
*The only
algebra over the Steenrod algebra of this type is B1;1(theorem 2.1).
For r|p - 1, the space Yk;rfits into the fibration Yk;1 ! Yk;r ! BZ=r. A sp*
*ectral
sequence argument establishes the isomorphisms
H*(Yk;r; Fp) ~=H*(Yk;r; Fp) ~=H*(Yk;1; Fp)Z=r ~=BZ=r1;1= B1;r:
For any map f : BA ! Y , A an abelian group, the connected group BA acts on *
*the
mapping space map (BA; Y )f. The Borel construction
Bor (Y; f) := EBA xBA map (BA; Y )f
sits in a sequence of fibrations
BA ! map (BA; Y )f ! Bor(Y; f) ! B2A:
26 J. AGUADE, C. BROTO AND D. NOTBOHM
Lemma 7.2. Let f : BA ! Y be a map, A a compact abelian group. Then, eY :
f *
* __
map (BA; Y )f '_Y_ if_and_only if there exists a principal fibration_BA!- Y !-*
* Y and
e__Y: map (BA; Y)c ' Y where c denotes the constant map. Moreover, Y ' Bor(Y; f*
*).
f *
* __
Proof. First let us assume that there exists a principal fibration BA!- Y !- *
* Y . By
`Thom-theory' ([26], revisited in [21]) this principal fibration establishes a *
*diagram of
principal fibrations
__
map (BA; BA)c ----! map (BA; Y )f----! map (BA; Y)c
? ? ?
(*) eBA?y eY?y e__Y?y
__
BA ----! Y ----! Y
The product h . g of two maps h : BA!- BA and g : BA!- Y is given by the acti*
*on of
BA on Y . In general the fiber in the top row consists of all maps h : BA!- BA*
* such that
h . f ' f,_in_particular it contains the component of the constant map. The fun*
*damental
group ss1(Y )_acts_on BA via maps homotopic to the identity. The fundamental g*
*roup
ss1(map (BA; Y)c) acts on the fiber via this action, which therefore also acts *
*via maps
homotopic to the identity. Because the total space of the fibration is connect*
*ed, this
action also must permute the components of the fiber, which is therefore connec*
*ted and
consists only of the component of the constant map. Moreover, because A is a co*
*mpact
abelian group the map eBA is an equivalence.
If e__Yalso is an equivalence, then eY is also an equivalence, which proves *
*one half of the
statement.
Now we assume_that map (BA; Y )f ' Y . The space BA acts on map (BA; Y )f w*
*ith
homotopy orbit Y := Bor(Y; f). This establishes the desired principal fibration
__
BA ----! map (BA; Y )f ' Y ----! Y :
Applying `Thom-theory' again, yields the diagram (*) of principal fibrations. T*
*his time the
first two vertical maps are equivalences_and so is the third one. Moreover, the*
* equivalence
of both rows in (*) proves that Y ' Bor(Y; f).
The following lemma may also be found in [20].
Lemma 7.3. Let K ! G ! H be an exact sequence of topological groups. If the
evaluation map map (BK; Y )c ! Y is an equivalence, then
a
map (BH; Y ) ! map (BG; Y )g
g|BK 'c
is an equivalence, where c indicates a constant map.
Proof. H acts on gBK := EG=K ' BK freely, and on Y trivially. The canonical *
*map
Y ! map (BK; Y )c is equivariant and an equivalence. Therefore
a
map (BH; Y ) ' Y hH ' (map (BgK ; Y )c)hH ' map (BG; Y )g:
g|BK 'const
HOMOTOPY CLASSIFICATION 27
Here Y hH denotes the homotopy fixed point set. The last equivalence follows fr*
*om [14].
Now we can state and prove the main result of this section. For j k + 1, le*
*t Yk(j)
denote the homotopy fibre of the composition
B2q
Xk!- B2Z=pk+1 --! B2Z=pk+1-j;
where q : Z=pk+1 ! Z=pk+1-j is the projection. These spaces fit into a sequence
Yk := Yk(0) ! Yk(1) ! . .!.Yk(k + 1) = Xk:
The realization BZ=p ! Xk of the composition H*(Xk; Fp) ! Fp[x] ! H*(BZ=p; Fp) *
*can
be lifted to a map BZ=pk+2 ! Yk.
Table 10.4 in the appendix displays the above sequences. In table 10.5 one c*
*an read the
cohomology algebras of the spaces involved in table 10.4.
Proposition 7.4. (1)
8
>>>B1;1; j = 0
< B0;1; j = 1
H*(Yk(j); Fp) ~=> 0
>>:B0;1; 2 j k
A01; j = k + 1:
(2) The spaces fit into fibrations
gj;l fj;j+l a2
BZ=pl ----! Yk(j) ----! Yk(j + l)----! B2Z=pl;
where l k - j + 1. The last map classifies the two dimensional class x*
* and is an
H2( ; Fp)-isomorphism. The first map is a realization of the composition
H*(Yk(j); Fp) ! Fp[x] ! H*(BZ=pl; Fp):
(3) For l k - j + 1, the evaluation e : map (BZ=pl; Yk(j))gj;l! Yk(j) is a*
* homotopy
equivalence. Moreover, Yk(j + l) ' Bor(Yk(j); gj;l).
(4) For l = k - j + 2, there is a fibration
BZ=pl-1 ! map (BZ=pl; Yk(j))gj;l! Ek ! B2Z=pl-1:
(5) There exists a map B2ss ' BS1bp! Yk, which is a realization of the comp*
*osition
H*(Yk; Fp) ! Fp[x] ! H*(BS1bp; Fp)
Proof. For j 1, a Serre spectral sequence argument for the fibrations
BZ=pk+1-j ! Yk(j) ! Xk ! B2Z=pk+1-j
28 J. AGUADE, C. BROTO AND D. NOTBOHM
shows that H*(Yk(j); Fp) is a P E-algebra of type (2,3). The action of the Stee*
*nrod algebra
will be calculated later.
(2) follows from the commutativity of the diagram
BZ=pl ----! BZ=pk+1-j ----! BZ=pk+1-j-l ----! B2Z=pl
flfl ? ? fl
fl ?y ?y flfl
BZ=pl ----! Yk(j) ----! Yk(j + l) ----! B2Z=pl
?? ? ? ?
y ?y ?y ?y
* ----! Xk ________ Xk ----! *
?? ? ? ?
y ?y ?y ?y
B2Z=pl ----! B2Z=pk+1-j ----! B2Z=pk+1-j-l ----! B3Z=pl
The conditions on the maps can be easily obtained by looking at the differentia*
*ls of the
associated Serre spectral sequences.
The classifying map Yk(j) ! B2Z=pj of the fibration Yk(0) ! Yk(j) is an H*( *
*; Fp)-
isomorphism in low dimensions. This proves that fi(x) = fi(y) = 0 for j 2 and*
* that
fi(x) = y for j = 1, which determines one part of the Steenrod algebra action.
(3) follows from (2) and lemma 7.2. We only have to show that map (BZ=pl; Yk*
*(j))c '
Yk(j) for all l and j. For l = 1 this is a consequence of theorem 3.4 and [17].*
* Now, lemma
7.3 and an induction over l proves the statement.
To prove (4), we use again lemma 7.3. In this case, it is l = k - j+ 2 and i*
* = k - j+ 1,
map (BZ=pk-j+2; Yk(k + 1))fj;k+1gj;k-j+2' map (BZ=p; Yk(k + 1))gk+1;1;
because fj;k+1gj;k+2|BZ=pk-j+1' c, Yk(k + 1) ' Xk and map (BZ=p; Xk)gk+1;1' Ek *
*(see
the proof of 6.1). We can apply the results of section 6 and get a principal fi*
*bration
map (BZ=pk-j+2; BZ=pk-j+1)q ! map (BZ= pk-j+2; Yk(j))gk-j+2;j
! map (BZ=p; Yk(k + 1))gk+1;1' Ek:
The first mapping space is equivalent to BZ=pk-j+1. This establishes the fibrat*
*ion of (4).
The composition
pk+1
B2ss ' BS1bp----! BS1bp ----! Ek ' map (BZ=p;Xk)gk+1;1 ----! Xk
can be lifted to BS1bp! Yk. Obviously, this map induces the desired map in mod*
* p
cohomology of (5).
To complete the proof of (1), we finally have to calculate P 1(y). For j = k*
* + 1, there is
nothing to show. P 1(y) 6= 0, for j k, contradicts the fact that map (BZ=p; Yk*
*(j))gj;1'
Yk(j), as theorem 3.1 shows.
HOMOTOPY CLASSIFICATION 29
To get a complete picture, we define X1 := S3bpand Y1 := S3<3>bp. Then,
BS1bp! Y1 ! X1 ! B2S1bp' (B2Z=p1 )bp
are fibrations. There exists a long sequence of maps
Y1 ! Y1 (1) ! Y1 (2) ! . .;.
pj
where Y1 (j) is the homotopy fibre of the map S3 -! B2S1bpof degree pj. Moreo*
*ver,
proposition 7.4 holds for Y1 . The proof is analogous.
Corollary 7.5. For every k 1 and every 0 j k, the homotopy type of Yk(j)
determines the homotopy type of every space in the sequence associated to Yk.
Proof. Proposition 7.4 (2) and (3).
Corollary 7.6. The spaces Yk;rare of different homotopy type.
Proof. For r = 1, this follows from proposition 7.4 (3) and (4). For r > 1, th*
*e map
Yk;1= Yk ! Yk;rinduces an equivalence Yk ' map (BZ=p; Yk)g0;1' map (BZ=p; Yk;r)*
*g,
g0;1
where g : BZ=p --! Yk!- Yk;r. This follows from theorem 3.1 and [17].
Next we construct a list of spaces realizing the algebras B0;r, for r|p - 1.*
* Let s 2 Z=r
(Z=p)* ~=Aut(Z=p) be a generator. The diagram
Z=p x Z=p ----! Z=p
? ?
sxs?y s?y
Z=p x Z=p ----! Z=p
commutes, because s is given by a multiplication. The horizontal arrows are gi*
*ven by
addition. Passing to classifying spaces and mapping spaces and taking adjoints*
* yields a
commutative diagram
BZ=p x map (BZ=p; Yk;r)g ----! map (BZ=p; Yk;r)g
? ?
sxmap(s;id)?y map(s;id)?y
BZ=p x map (BZ=p; Yk;r)g ----! map (BZ=p; Yk;r)g:
g0;1
Here, g denotes the composition BZ=p --! Yk!- Yk;r. Because H*(Yk;r; Fp) is *
*a P E-
algebra of type (2pr; 2pr + 1), the component of g is fixed by the Z=r-action. *
* Thus, we
get a Z=r-action on the quotient Bor(Yk;r; g) ' Bor(Yk; g0;1) ' Yk(1), k 0. T*
*he first
equivalence follows from theorem 3.1 and [17] and the second equivalence is fro*
*m lemma
7.2.
Now, we define Z0k;r:= EZ=r xZ=rYk(1) and Zk;r:= (Z0k;r)bp. We also put Zk =*
* Zk;1.
30 J. AGUADE, C. BROTO AND D. NOTBOHM
Proposition 7.7. For 0 < k 1 and r|p - 1, all the spaces Zk;r are pairwise not
homotopy equivalent, and H*(Zk;r; Fp) ~=B0;ras algebras over the Steenrod algeb*
*ra.
Proof. For r = 1, Zk;r' Yk(1), and H*(Yk(1); Fp) ~= B0;1by proposition 7.4 (1).*
* For
r > 1, the calculation of H*(Zk;r; Fp), is analogous to the calculation of H*(Y*
*k;r; Fp) in the
proof of proposition 7.1. Let f : BZ=p ! Zk;1! Zk;rbe the obvious composition.*
* By
theorem 3.1 and [17],
map (BZ=p; Zk;r)f ' Zk;1' Yk(1):
Now, the statement follows from corollary 7.5.
8. Classification of spaces realizing Bi;r. In this section we classify up to p*
*-completion
the possible homotopy types of spaces realizing Bi;r. By theorem 4.3 we only h*
*ave to
consider the cases i = 0; 1. Let Yk;rand Zk;rbe the spaces constructed in secti*
*on 7 with
H*(Yk;r; Fp)~=B1;r
H*(Zk;r; Fp)~=B0;r
We will show that these spaces form a complete list of p-complete homotopy type*
*s realizing
B1;rand B0;r, respectively.
The next proposition is an immediate consequence of well know properties of *
*the Bock-
stein spectral sequence (see section 5).
Proposition 8.1. Let X be a space with p-adic cohomology of finite type over ^Z*
*p. If
H*(X; Fp) = Bi;rthen in the ^Zp-cohomology Bockstein spectral sequence {Bl; dl}*
* for X
we have B1 = 0.
Theorem 8.2. (1) If H*(X; Fp) ~=B1;rthen there exists 0 k 1 such that ^Xp' Y*
*k;r.
(2) If H*(X; Fp) ~=B0;rthen there exists 0 < k 1 such that X^p' Zk;r.
Proof. (cf. tables 10.4 and 10.5.) Let Y be the p-completion of a space X reali*
*zing B1;r.
Then Y is 1-connected, p-complete and realizes also B1;r. By 5.7 the p-adic coh*
*omology of
Y is of finite type over ^Zpand by 8.1 and 5.7 all homotopy groups of Y are fin*
*ite p-groups.
Consider first the case of B1;1. We will construct a sequence of maps
Y := Y (0) ! Y (1) ! Y (2) ! . . .;
such that H*(Y (j); Fp) ~=Fp[aj] E(bj) isomorphic to either A(j-1)1or B(j-1)0;*
*1, j 1, and
aj
such that there exists a fibration sequence BZ=pj!- Y (0)!- Y (j) -! B2Z=pj, *
*where the
last map is algebraically given as in Proposition 7.4 (2).
Let us assume that we already constructed Y (j), j 1, with H*(Y (j); Fp) ~=*
*Fp[aj]
E(bj) isomorphic to either A(j-1)1or B(j-1)0;1. Let gj;1: BZ=p ! Y (j) be the *
*realization
of the composition H*(Y (j); Fp) ! Fp[aj] ! H*(BZ=p; Fp). Then the computation *
*of the
T functor on the algebras A and B in 3.1 and 3.5 and the results of [17] imply *
*that a
necessary and sufficient condition for
map (BZ=p; Y (j))fj' Y (j)
HOMOTOPY CLASSIFICATION 31
is P 1(bj) = 0, i.e. H*(Y (j); Fp) = B(j-1)0;1. Hence, for P 1(bj) = 0, there e*
*xists a principal
fibration (lemma 7.2)
BZ=p ! Y (j) ! Bor(Y (j); fj) =: Y (j + 1) ! B2Z=p;
which fits into the commutative diagram
* ----! Y (0) ________ Y (0) ----! *
?? ? ? ?
y ?y ?y ?y
fj
BZ=p ----! Y (j) ----! Y (j + 1)----! B2Z=p
flfl ? ? fl
fl aj?y aj+1?y flfl
ajfj
BZ=p ----! B2Z=pj ----! B2Z=pj+1 ----! B2Z=p :
A short calculation shows that Bor(B2Z=pj; ajfj) ' B2Z=pj+1. This establishes t*
*he map
aj+1. The differentials in the Serre spectral sequence of the fibration given *
*by the right
three terms in the middle row are given by the equations d2(aj) = fi() and d2(b*
*j) =
0. The equations follow from a comparison with the spectral sequence of the fi*
*bration
in the bottom row. Now a straightforward calculation shows that H*(Y (j + 1); *
*Fp) ~=
Fp[aj+1] E(bj+1) is a P E-algebra of type (2; 3) with the relation fij+1(aj+1)*
* = bj+1; i.e.
H*(Y (j +1); Fp) is isomorphic to either A(j)1or B(j)0;1. The relation on the B*
*ockstein follows
from the fact H2(Y (j + 1); Z) ~=ss2(Y (j + 1)) ~=ss2(B2Z=pj+1) ~=Z=pj+1. This*
* finishes
the induction step.
The construction of Y (1) does not fit into this picture, but it is done in *
*the obvious way
by starting with a map BZ=p ! Y (0) which also is algebraically given as in pro*
*position
7.4.
We can continue as long as P 1(bj) = 0. Let us first assume that we can cons*
*truct only a
finite sequence of spaces and let Y (k + 1) denote the last space. Then H*(Y (k*
* + 1); Fp) ~=
Fp[x] E(y) is a P E-algebra of type (2; 3) such that fi(k+1)(x) = y and P 1(y)*
* 6= 0.
Since P 2(y) = 0 by unstability, the only possibility for P 1(y) is P 1(y) = xp*
*-1y and so
H*(Y (k + 1); Fp) ~=A(k)1. This implies that Y (k + 1) ' Xk (theorem 6.1) and Y*
* = Y (0) '
Yk = Yk;1(corollary 7.5).
If the sequence is infinite, we define Y (1) := hocolimY (j). In the Milnor *
*sequence
1!- lim-1H*+1(Y (j); Fp)!- H*(Y (1); Fp)!- lim-H*(Y (j); Fp)!- 1 ;
the first term vanishes because all the groups are finite, and
lim-H*(Y (j); Fp) ~=H*(S3; Fp):
Hence Y (1)bp' S3bp. Let F be the homotopy fibre of Y (0) ! Y (1). Since the *
*direct
limit of a directed system of fibrations is again a fibration, the fibration F *
*! Y (0) ! Y (1)
is the direct limit of the fibrations
BZ=pj ! Y (0) ! Y (j):
32 J. AGUADE, C. BROTO AND D. NOTBOHM
Hence, F ' lim-!jBZ=pj = BZp1 and by taking the p-completion we obtain a fibrat*
*ion
BS1bp-! Y (0)!- S3bp
classified by a map S3bp-! B Aut(BS1bp) into the classifying space of the monoi*
*d of self
homotopy equivalences of BS1bp([23]), which lifts to a map
S3bp-! BS Aut(BS1bp) ' B2S1bp' K(ss; 3) :
where S Aut denotes the self homotopy equivalences which are homotopic to the i*
*dentity.
This map is classified by degree and the p adic units are the only possible one*
*s that produce
the right cohomology of Y (0) and therefore Y = Y (0) ' Y1 .
Now let Y 0be the p-completion of a space realizing B1;r. Let f : BZ=p!- Y*
* 0be a
realization of the composition B1;r-! B1;1-! H*(BZ=p; Fp). By the computation o*
*f the
T functor in theorem 3.1 and the results of [17], map (BZ=p; Y 0)f is a realiza*
*tion of B1;1.
Thus, there exists an equivalence h : map (BZ=p; Y 0)f ' Yk for some 0 k 1. T*
*he space
map (BZ=p; Y 0)f inherits a Z=r-action from the Z=r-action on BZ=p. The compone*
*nt of
f is fixed under this action because H*(Y 0; Fp) ~=B1;r. h induces an equivaria*
*nt map in
H*( ; Fp) because the canonical map BZ=p ! map (BZ=p; Y 0)f is equivariant. By *
*lemma
8.3 below, we can replace h by an equivariant equivalence. Taking homotopy orbi*
*ts gives
equivalences
Y 0' (EZ=r xZ=rmap (BZ=p; Y 0)f)bp' (EZ=r xZ=rYk)bp' Yk;r:
This finishes the proof of part (1).
To prove (2), let Z be the p-completion of a space X realizing the algebra B*
*0;1. The
homotopy fibre of the classifying map x : Z!- B2Z=p of the 2-dimensional class*
* x is a
realization of B1;1and hence, equivalent to some Yk, k 1. By corollary 7.5 it*
* follows
that Z ' Zk.
If the p-complete space Z0 realizes the algebra B0;r, we can proceed as in t*
*he case of
B1;r. We have an equivalence map (BZ=p; Z0)f ' Zk for some k and for a suitabl*
*e map
f : BZ=p!- Z0. Now all the above arguments go through with minor changes. This*
* shows
that Z0 ' Zk;rand finishes the proof.
Lemma 8.3. (1) Let Y be a space equipped with a Z=r action, and let h : Y ! *
*Yk
be an equivalence, such that H*(h; Fp) is equivariant. Then, there exists an e*
*quivalence
h0: Y ! Yk, which is equivariant.
(2) Let Z be a space equipped with a Z=r action, and let h : Z ! Zk be an e*
*quivalence,
such that H*(h; Fp) is equivariant. Then, there exists an equivalence h*
*0: Z ! Zk,
which is equivariant.
Proof. We only prove (1), the proof of (2) is analogous. There exists a map g :*
* BZ=pk!- Y
and an equivalence f : Y ' hofib(a : Bor (Y; g) ! B2Z=pk), where hofib denotes*
* the
homotopy fibre. Analogously to the construction of the spaces Yk;r, the Z=r act*
*ion on Y
passes to Bor(Y; f) and the map a is equivariant up to homotopy. Z=r acts on Z=*
*pk via the
HOMOTOPY CLASSIFICATION 33
inclusion Z=r Z=p- 1 ^Zp*. By [27] (see lemma 6.5), we can replace a by an eq*
*uivariant
map. This induces a Z=r-action on the homotopy fibre. The homotopy equivalenc*
*e f is
equivariant up to homotopy. This construction can be applied to Yk as well.
The Borel construction yields an equivalence Bor(Y; g) ' Bor(Yk; g0;k) which*
* is also
equivariant in mod-p cohomology. Both spaces are equivalent to Xk. As in lemm*
*a 6.5
we can replace this equivalence by an equivariant equivalence. Taking homotopy*
* fibers
produces an equivalence Y ! Yk, which is equivariant up to homotopy. Again, the*
* Wo-
jtkowiak argument establishes an equivalence Y ! Yk which is equivariant.
9. Homotopy properties of the constructed spaces.
In [9], for any map f : A ! B between spaces, Dror Farjoun constructed a loc*
*alisation
functor
Lf : Spaces ! Spaces :
Here, Spaces means the category of topological spaces, of CW -complexes, or the*
* simplicial
category. In this section we will, among other things, compute the value of th*
*is functor
when applied to some of the spaces constructed in the previous sections, in the*
* particular
case in which f is the map BZ=p ! *. The functor Lf is coaugmented, homotopica*
*lly
idempotent, and takes values among the f-local spaces. A space Y is called f-lo*
*cal, if the
map
f*: map(B; Y ) ! map (A; Y )
is a homotopy equivalence. Moreover, the coaugmentation l : X ! LfX into the lo*
*cali-
sation LfX is homotopically universal, i.e. for any map X ! Z into a f-local sp*
*ace Z,
there exists a map LfX ! Z, unique up to homotopy, such that
X ______X
? ?
l?y ?y
LfX --! Z
commutes up to homotopy. Actually, l induces a homotopy equivalence
'
l*: map(LfX; Z) -! map (X; Z)
for any f-local space Z. Such functors satisfy several properties by general n*
*onsense
arguments; e.g we have
Lemma 9.1. Let g : X ! Y be a map between spaces, then the following statements*
* are
equivalent:
(1) g induces a homotopy equivalence LfX ' LfY .
(2) For any f-local space Z, g*: [Y; Z]!- [X; Z] is a bijection.
(3) For any f-local space Z, g*: map(Y; Z)!- map (X; Z) is a homotopy equ*
*iva-
lence.
34 J. AGUADE, C. BROTO AND D. NOTBOHM
Lemma 9.2. (1) For any small category C and for any functor F : C ! Spaces,
Lf(hocolimC F ) ' Lf(hocolimC(Lf O F )) :
(2) The homotopy inverse limit over any small category of f-local spaces is*
* f-local.
Proof. For an f-local space Z, the map map (hocolimC Lf O F; Z)!- map (hocolim*
*C F; Z)
can be factored as
map(hocolimC Lf O F; Z) ' holimCmap (Lf O F; Z) '
' holimCmap (F; Z) ' map (hocolimC F; Z):
Then, (1) is a consequence of 9.1. The proof of statement (2) is similar.
We will also use some facts about Lf that we quote from [11]:
Lemma 9.3. Assume that F!- E!- B is a fibration.
(1) If LfF ' * then LfE ' LfB.
(2) For f: W ! *, if B is f-local, then Lf preserves the fibration.
For a space W , we denote the localisation with respect to the map W ! * by *
*LW . Then,
a space X is W -local if and only if map (W; X) ' X or equivalently, for X conn*
*ected, if
and only if map *(W; X) ' *. In this section we are interested in the localisa*
*tion with
respect to BZ=p.
Some elementary calculations of LBZ=p are provided by the next two results.
Lemma 9.4. Let ss denote a discrete group,
(1) K(ss; 1) is BZ=p-local if and only if ss is p-torsion free.
(2) K(ss; n) is BZ=p-local for all n 1 if and only if ss is a uniquely p-d*
*ivisible abelian.
(3) If ss is a p-group then LBZ=p(K(ss; n)) ' * if n > 1 or ss is finite.
Proof. In general a direct computation of homotopy groups shows that the connec*
*ted
component containing the constant map of map (BZ=p; K(ss; 1)) is homotopy equiv*
*alent to
K(ss; 1). Now the set of components of map (BZ=p; K(ss; 1)) is Rep(Z=p; ss), he*
*nce there is
a unique component if and only if ss is p-torsion free. This proves (1).
From a computation of homotopy groups it follows that K(ss; n), n 2 is BZ=p*
*-local if
and only if Hr(Z=p; ss) = 0 for 1 r n, that is, if and only if ss is uniquely*
* p divisible.
This is (2).
Finally we prove (3). Clearly LBZ=p(BZ=p) ' *, then we use induction on the *
*order of
ss and n in order to get the result for any finite p-group. A general p-group i*
*s direct limit
of its finite subgroups, hence the result follows by 9.2(1).
Remark 9.5. The following explicit calculations will be useful later.
(1) For any n 2, LBZ=pK(Z; n) ' K(Z[_1p]; n), and
(2) For any n 2, LBZ=pK(^Zp; n) ' K(Q^p; n).
This is computed using the exact sequences 0 ! Z ! Z[_1p] ! Z=p1 ! 0 and 0 ! ^*
*Zp!
Q^p! Z=p1 ! 0 and then applying 9.3 and 9.4.
Examples of BZ=p-local spaces are provided by the Sullivan conjecture:
HOMOTOPY CLASSIFICATION 35
Lemma 9.6.
(1) Any finite CW-complex is BZ=p-local. (This is the Sullivan conjecture: *
*[19].)
(2) Let X be a connected nilpotent space with H*(X; Fp) of finite type, the*
*n X is
BZ=p-local if and only if H*(X; Fp) is locally finite as module over th*
*e Steenrod
algebra. ([18])
The next lemma lists several properties of BZ=p-local spaces.
Lemma 9.7.
(1) If Z is BZ=p-local, then Z is BZ=pk-local for every 1 k 1. If, in add*
*ition, Z
is p-complete, then Z is also BS1-local.
(2) Let Z be a connected space and eZ! Z the universal covering. If Z is B*
*Z=p-
local, then eZis also BZ=p-local. Reciprocally, if ss1(Z) is p-torsion *
*free and eZis
BZ=p-local, then Z is also BZ=p-local.
The proof is based on the following lemma of Zabrodsky [28] (see also [19]).
Lemma 9.8. Let G be a toplogical group and G ! E ! B a principal fibration. If,*
* for
a space X, map (G; X)const' X, then
map (B; X) ' map (E; X)f|G'const:
The mapping space map (E; X)f|G'constconsists of the components of all maps *
*f : E !
X, whose restriction f|G is homotopic to the constant map.
Proof of lemma 9.7. (1). The principal fibration BZ=p ! BZ=pk+1 ! BZ=pk, the Za*
*brod-
sky lemma and an induction prove (1) for k < 1. Moreover, for a BZ=p-local spac*
*e Z, the
canonical map map (BZ=pk+1; Z) ! map (BZ=pk; Z) is a homotopy equivalence. The*
*re-
fore,
map (BZ=p1 ; Z) ' lim-map(BZ=pk; Z) ' lim-Z ' Z ;
k k
and Z is BZ=p1 -local. If Z is also p-complete map (BS1; Z) ' map (BZ=p1 ; Z) *
*' Z,
which shows that Z is BS1-local and finishes the proof of (1).
(2). Assume that Z is connected and BZ=p-local. Let Z ! K := K(ss1(Z); 1) be*
* the
classifying map of the universal covering eZ ! Z. Applying the functor map (BZ*
*=p; )
establishes a commutative diagram of fibrations
map(BZ=p; eZ)--! map(BZ=p; Z) --! map (BZ=p; K)const
? ? ?
eeZ?y eZ?y eK?y
Ze --! Z --! K :
The vertical arrows are given by the evaluation. In the upper middle term we d*
*o not
have to consider particular components, because Z is BZ=p local; i.e. there is*
* only the
component of the constant map and eZ is a homotopy equivalence. Since eK is a*
*lso a
homotopy equivalence, this is also true for eZe. That is to say that eZis BZ=p-*
*local.
36 J. AGUADE, C. BROTO AND D. NOTBOHM
Finally, if ss1(Z) is p-torsion free, Bss1(Z) is BZ=p-local by 9.4(1) and th*
*en, according
to 9.3(2) Z is BZ=p-local if and only if eZis BZ=p-local. This finishes the pr*
*oof of the
second statement.
Now we are prepared to start with the calculation of the localisations of th*
*e spaces we
constructed in the previous sections.
The next result is actually a particular case of a more general result of Ne*
*isendorfer.
Lemma 9.9. S3, LBZ=pS3<3> and LBZ=p(S3<3>bp) are homotopy equivalent after comp*
*le-
tion.
Proof. By lemma 9.6, S3 and S3bpare BZ=p-local. So, by lemma 9.3 LBZ=p preserve*
*s the
fibration BS1 ! S3<3> ! S3 as well as its p-completion, hence, by lemma 9.5 we *
*obtain
fibrations:
K(Z[1_p]; 2) ! LBZ=pS3<3> ! S3
and
K(Q^p; 2) ! LBZ=p(S3<3>bp) ! S3bp
and the p-completion of those gives the result.
Recall that in section 5 the space Ek was defined as the total space of cert*
*ain fibration
BS1bp! Ek ! S1bp
that has a section s: S1bp! Ek. Then E0kis the homotopy cofibre of this section*
*. Finally,
Xk was defined as the p completion of E0k. For the localization of that spaces *
*we obtain:
Lemma 9.10. (1) (LBZ=pEk)bp' S1bp.
(2) (LBZ=pE0k)bp' *.
(3) LBZ=pXk ' *.
Proof. Since S1bpis BZ=p-local, LBZ=p preserves the above fibration and we obta*
*in a fi-
bration
K(Q^p; 2) ! LBZ=pEk ! S1bp:
The p-completion of this fibration proves (1).
This fibration has also a section s: S1bp! LBZ=pEk. Let C be the homotopy co*
*fibre of
this section. Then C is simply connected and mod p acyclic, hence BZ=p-local by*
* lemma
9.6. Now, since a homotopy cofibre is a special sort of homotopy colimit, by 9.*
*2 we obtain
LBZ=pE0k' LBZ=pC ' C
and therefore (LBZ=pE0k)bp' *. This is statement (2).
According to the next lemma, (3) follows from (2) because Xk is 1-connected *
*and
Hi(Xk; ^Zp) is finite for all i > 0 (see 5.5 and 5.8.)
HOMOTOPY CLASSIFICATION 37
Lemma 9.11. Let X be a space for which X^pis 1-connected. If Hi(X^p; ^Zp) is fi*
*nite for
all i > 0 and (LBZ=pX)bp' *, then LBZ=p(X^p) ' *.
Proof. We want to show that for any BZ=p-local space Z, any map X^p ! Z factors
through a point. that is [X^p; Z] = * for any connected BZ=p-local space Z.
If Z is BZ=p-local, the universal covering Ze, is also BZ=p-local and since *
*X^p is 1-
connected ffi
[X^p; Z] ~=[X^p; eZ] ss1(Z)
hence it is enough to show that [X^p; Z] = * for all Z which is 1-connected and*
* BZ=p-local.
If Z is 1-connected and BZ=p-local, so is ^Zp. Since Hi(X^p; ^Zp) if finite *
*for all i > 0 and
X^p is 1-connected, 5.7 implies that the homotopy groups of X^pare finite p-gro*
*ups. Then,
the arithmetic fracture lemma shows that [X^p; ^Zp] ~=[X^p; Z] and
[X^p; Z] ~=[X^p; ^Zp] ~=[LBZ=pX; ^Zp] ~=[(LBZ=pX)bp; ^Zp] ~=[*; ^Zp] = *
The spaces Yk(j) for k 1 and 0 j k + 1, were constructed out of Xk in sec*
*tion 7
in such a way that they fit in sequences of fibrations with fibre BZ=p:
Yk(0) ! Yk(1) ! : :!:Yk(k + 1) = Xk:
Here Yk = Yk(0) is the kth fake S3<3> and Y1 = S3<3>bpalso fits in one such se*
*quence
Y1 ! Y1 (1) ! Y1 (2) ! : :w:ith Y1 (1) = hocolimjY1 (j) and Y1 (1)bp' S3bp.
Theorem 9.12. For all 0 j k + 1 1,
aeY (1) for k = 1
LBZ=pYk(j) ' LBZ=pYk(0) = LBZ=pYk ' 1
* for k < 1
Remark. Compare with 9.9 for k = 1.
Proof. The principal fibrations BZ=p ! Yk(j) ! Yk(j + 1) and lemma 9.2 establish
equivalences LBZ=pYk(j) ' LBZ=pYk(j + 1).
Therefore, if k < 1 we have LBZ=pYk(j) ' LBZ=pXk ' * by 9.10.
For k = 1 we have first that Y1 (1) is 1-connected and its mod p cohomology *
*is
finite hence it is BZ=p-local by 9.6(2). Then the result follows from 9.1 becau*
*se for any
BZ=p-local space Z the map Y1 (j) ! hocolimjY1 (j) = Y1 (1) induces
' '
map(Y1 (1); Z) -! holimjmap (Y1 (j); Z) -!
' '
-! holimjmap (LBZ=pY1 (j); Z) -! map (LBZ=pY1 (j); Z) :
38 J. AGUADE, C. BROTO AND D. NOTBOHM
Corollary 9.13. If k 6= 1, the l-fold suspensions of Y1 (j) and of Yk(j) are no*
*t homotopy
equivalent for all l.
Proof. By theorem 9.12, lLBZ=pY1 (j) ' lY1 (1) and this is 1-connected with fin*
*ite
mod p cohomology hence BZ=p-local. The suspensions are homotopy colimits. By le*
*mma
9.2 we have
LBZ=plYk(j) ' LBZ=plLBZ=pYk(j) ' LBZ=plY1 (1) ' lY1 (1) :
But for k < 1, the same argument proves (LBZ=plYk(j)) ' *.
Similiar arguments as in the proof of theorem 9.5 show that LBZ=p(Y1 (j)xZ=r*
*EZ=r) '
Y1 (1) xZ=rEZ=r, where Z=r acts canonically on Y1 (1). This space is mod-p acyc*
*lic
and using lemma 9.11 we deduce that LBZ=pY1;r ' *. For k < 1, one also can prov*
*e that
LBZ=pYk;r' *. Thus, the above application of the localisation functor does not *
*see any
difference between the suspensions of the different realisations of Bi;r. But t*
*here is a way
to distinguish between these spaces.
Let g : BZ=p ! Yk;rbe the map of section 7, and let h : YkxBZ=p ! Yk;rbe the*
* adjoint
of the equivalence Yk ' map (BZ=p; Yk;r)g. Z=(p - 1) acts on Yk. For every a 2 *
*Z=p* we
have a map
axid lh
lYk x BZ=p ---! lYk x BZ=p --! l(Yk x BZ=p) --! lYk;r:
which has as adjoint a map
fa : lYk --! map (BZ=p; lYk;r) :
If a and b differ by a r-th power, the two associated maps fa and fb are homoto*
*pic,
because Yk;ris the homotopy orbit of the Z=r-action on Yk. There is also an obv*
*ious map
lYk;r! map (BZ=p; lYk;r), which is the standard section of the evaluation.
Let s = (p - 1)=r. Then, Z=s Z=p* consists of the congruence classes modulo*
* r-th
powers. All these maps together fit into a map
_
f : lYk;r_ lYk ! map (BZ=p; lYk;r) :
a2Z=s
Theorem 9.14. The map
_
f : lYk;r_ lYk ! map (BZ=p; lYk;r)
a2Z=s
is a homotopy equivalence.
Proof. We have to calculate the mapping space using the T -functor The T -funct*
*or is exact
and commutes with suspensions. Hence,
T (H*(lYk;r; Fp))~=T (He*(lYk;r; Fp) T (Z=p)
~=lT (H*(lYk;r; Fp)) Z=p
~=( M lH*(Yk; Fp)) lH*(Yk;r; Fp) Z=p
a2Z=s
~=H*(( _ lYk) _ lYk;r; Fp) :
a2Z=s
HOMOTOPY CLASSIFICATION 39
All the isomorphism are obvious, but the second last. This one follows from the*
*orem 3.1,
the identity TconstH*(Yk;r; Z=p) ~=H*(Yk;r; Z=p), the fact that every map H*(Yk*
*;r; Fp) !
H*(BZ=p; Fp) factors over H*(Yk;; Z=p) and that every two factorizations differ*
* by an r-th
power. By construction, this series of isomorphisms is just the map inducedWby *
*f, which
shows that f is a mod-p equivalence [17]. The integral homology of ( a2Z=slYk)*
*_lYk;r
consists of finite p-torsion in each dimension. Moreover, the spaceWis 3-conect*
*ed. The mod C
Hurewicz theorem for the class of finite p-groups shows that ss*((W a2Z=slYk) _*
* lYk;r)
consists of finite p-torsion in each dimension. Hence, by [6] ( a2Z=slYk) _ l*
*Yk;ris
p-complete. Because Yk;ris p-complete, f is an equivalence ([17]).
Corollary 9.15. If k 6= 1, the l-fold suspensions lY1;r and lYk;rare not homoto*
*py
equivalent for all l.
Proof. Applying localisation to the mapping space gives
_
LBZ=p(map (BZ=p; lYk;r)) ' LBZ=p(lYk;r_ lYk)
a2Z=s
_
' LBZ=p(LBZ=p(lYk;r) _ LBZ=p(lYk))
a2Z=s
_
' LBZ=p(lLBZ=p(Yk;r) _ lLBZ=p(Yk)) :
a2Z=s
The last two equivalences follow from lemma 9.1 and lemma 9.2. By proposition *
*9.13
and the following remarks, for k = 1 and k < 1, these spaces cannot be homotopy
equivalent.
Remark 9.16. Using the same methods and ideas, one can also distinguish between*
* the
l-fold suspensions of Zk;r, k < 1, and Z1;r.
Finally, we discuss the question of which of these spaces are H-spaces. The*
* spaces
Y1 = S3<3>bpand Z1 = Y1 (1) are loop spaces, in particuliar H-spaces. This f*
*ollows
easily from the construction. But these are the only ones among the spaces Yk;r*
*and Zk;r
which are H-spaces, as the following proposition shows.
Proposition 9.17. For k < 1 or r > 1, the spaces Yk;rand Zk;rcannot carry an H-*
*space
structure.
Proof. If the algebras B0;rand B1;rhave the structure of a Hopf algebra, they a*
*re prim-
itively generated. For r > 1, the Steenrod power P pi, i = 0; 1, maps the primi*
*tive 2pir-
dimensional class on a nonprimitive class. Hence, B0;rand B1;rare Hopf algebra*
*s only
for r = 1.
40 J. AGUADE, C. BROTO AND D. NOTBOHM
Now, we assume that Yk(j), 0 j k, is an H-space. We consider the diagram
BZ=p x BZ=p ----! BZ=p
? ?
gj;1xgj;1?y gj;1?y
(3) Yk(j) x Yk(j) ----! Yk(j)
?? ?
y fj;j+1?y
Yk(j + 1) x Yk(j + 1) Yk(j + 1)
where denotes the multiplication. The upper square commutes in mod-p cohomolo*
*gy,
because in H*(Yk(j); Fp) the 2-dimensional class for j 1 and the 2p-dimensiona*
*l class
for j = 0 are primitive. Thus, the upper square commutes up to homotopy. The ob*
*vious
composition fj;j+1(gj;1x gj;1) : BZ=p x BZ=p ! Yk(j + 1) is homotpic to the con*
*stant
map and, by theorem 3.1 and taking the adjoint,
map (BZ=p x BZ=p; Yk(j + 1))c' map (BZ=p; map(BZ=p; Yk(j + 1))c)c
' map (BZ=p; Yk(j + 1))c
' Yk(j + 1):
Both vertical columns in (3) are principal fibrations. We can apply lemma 9.3,*
* which
establishes a map
: Yk(j + 1) x Yk(j + 1) ! Yk(j + 1)
making the lower square commutative up to homotopy. As easily shown, is an H-s*
*pace
structure on Yk(j + 1).
If Yk = Yk(0) is an H-space, the above induction procedure shows that Yk(k +*
* 1) = Xk
carries also an H-space structure. But this is a contradiction, because the Ste*
*enrod power
P 1maps the `primitive' 3-dimensional class of H*(Xk; Fp) onto a nonprimitive c*
*lass.
10. Appendix. Through this paper we have introduced several families of algebr*
*as
over the Steenrod algebra Ar, Bi;r, Cr, etc. as well as several families of sp*
*aces Ek(r),
Xk(r), Yk;r, etc. We think that the reader would find helpful to have the defi*
*nitions of
these algebras and spaces and the relationships between them displayed in a set*
* of tables.
In this appendix we include the following tables: Table 10.1 contains the defi*
*nitions of
the algebras over the Steenrod algebra introduced in section 2. Table 10.2 con*
*tains the
definitions and the mod p cohomology of the spaces introduced in sections 5 and*
* 7. In
table 10.3 we list some fibrations between these spaces. Tables 10.4 and 10.5 d*
*isplay the
spaces used in the proof of theorem 8.2 and their cohomology algebras.
HOMOTOPY CLASSIFICATION 41
____________________________________________________________________________||||
| (0) | deg(x) = 2r | fi(x) = y |
| Ar = Ar = Fp[x] E(y) | | 1 s |
||__________________________||deg(y)_=_2r_+_1___||P_(y)_=_rx_y______________|||*
*|||
| B = B(0)= F [x] E(y) | deg(x) = 2pir | fi(x) = y |
| i;r i;r p | | i |
||__________________________||deg(y)_=_2pir_+_1_||P_p(y)_=_(r_-_1)xsy_______|||*
*|||
| (0) | deg(x) = 2r | fi(x) = xz |
| Cr = Cr = Fp[x] E(z) | | |
||__________________________||deg(z)_=_1________||__________________________|||*
*||
||A0r___|same|as_Ar_but_with_fi(x)_=_fi(y)_=_0______________________________|||*
*||
0 same as B but with fi(x) = fi(y) = 0
||Bi;r__||________i;r_______________________________________________________|||*
*||
||C0r___|same|as_Cr_but_with_fi(x)_=_fi(z)_=_0______________________________|||*
*||
0 as algebras over the Steenrod algebra but with fi (x) = 0
| A(k)r |equal to Ar (j) |
||______|for|j__k_and_fi(k+1)(x)_=_y________________________________________|||*
*||
| B(k) |equal to B0i;ras algebras over the Steenrod algebra but with fi(j)(x)|*
*= 0
| i;r | |
||______|for|j__k_and_fi(k+1)(x)_=_y________________________________________|||*
*||
0 as algebras over the Steenrod algebra but with fi (x) = 0
| C(k)r |equal to Cr (j) |
||______|for|j__k_and_fi(k+1)(x)_=_xz_______________________________________||
Table 10.1
42 J. AGUADE, C. BROTO AND D. NOTBOHM
____________________________________________________________________________*
*___________________||||
(k)
|||||||Ek|=|EG xssB2ssOEk, k 0 ||H*(Ek; Fp) ~=C1 , (5.1) *
* ||
||||||| ffi | * ~ (k) *
* |
E |||||||Ek(r)|=|Ek Z=r, k 0, r|p - 1 ||H (Ek(r); Fp) = Cr , (5.3*
*) ||
(k)
|||||||E0k(r)|=|Cofibre(Bss_!_Ek(r)),_k__0,_r|p_-|1H*(E0k(r);|Fp)_~=Ar______*
*________||_________
||||||| 0 | *
* |
|||||||Xk(r)|=|(Ek(r))bp, k 0, r|p - 1 || *
* ||
(k)
X |||||||Xk|=|Xk(1),3k 0 ||H*(Xk(r); Fp) ~=Ar , (5.5*
*) ||
|||||||X1||~=(S_)bp_____________________________||__________________________*
*________||_________
||||||| 2 k+1 | *
* |
|||||||Yk|=|Fibre(Xk3! B Z=p ), k 0 || * *
* ||
|||||||Y1||= (S <3>)bp ||H (Yk;r; Fp) ~=B1;r, (7.1)*
* ||
|||||||Yk;r=|(EZ=r|xZ=rYk))bp,|0| k 1, r|p - 1 ||| 8 *
* |||
||||||| | >>B1;1; j*
* = 0 |
||||||| 2 k+1-j | >< *
* |
|||||||Yk(j) = Fibre(Xk ! B Z=p ) | * ~ B0;1; j*
* = 1 |
Y ||||||| k 0; 0 j k + 1 | H (Yk(j); Fp) = 0 *
* |
||||||| | >>>B0;1; 2*
* j k |
||||||| | : 0 *
* |
||||||| j | A1; j*
* = k + 1|
|||||||Y (j) = Fibre(S3 -p!(B2S1) ), j 0 | (7.4) *
* |
||||||||1| bp || *
* ||
|||||||Yk(0)|=|Yk;1= Yk || *
* ||
|||||||Yk(k|+|1)_=_Xk___________________________||__________________________*
*________||_________
||||||| | * *
* |
|||||||Zk;r= (EZ=r xZ=rYk(1))bp, 0 < k 1, r|p -|1H (Zk;r; Fp) ~=B0;r, (7.7)*
* |
Z ||||||||| || *
* ||
|||||||Zk|=|Zk;1=_Yk(1),_0_<_k__1_______________||__________________________*
*________||_________
Table 10.2
_______________________________________________
| B2ss ! Ek ! Bss |
| |
| Ek ! Ek(r) ! BZ=r |
| 2 k+1 |
| Yk ! Xk ! B Z=p |
| 2 |
| Yk ! Zk ! B Z=p |
| 2 k |
| Zk ! Xk ! B Z=p |
| 2 l |
| Yk(j) ! Yk(j + l) ! B Z=p ; l k - j + 1|
| 2 k+1-j |
| Yk(j) ! Xk ! B Z=p ; j 1 |
| Y ! X ! (B2S1) |
||____________1_____1__________bp_____________||_
Table 10.3
HOMOTOPY CLASSIFICATION 43
____________________________________________________________________________*
*__________________||
3<3> = Y ! Y (1) = Z ! Y (2) ! Y (3) ! . . . . .*
*!.S3
S ||||||bp1 1 1 1 1 *
* bp |
|||||| . *
* |
|||||| . *
* |
|||||| . *
* |
|||||| . *
* |
|||||| Yk ! Yk(1) = Zk ! Yk(2) ! Yk(3) ! . . . ! Yk(k + 1) = X*
*k |
|||||| . *
* |
|||||| . *
* |
|||||| . *
* |
|||||| Y3 ! Y3(1) = Z3 ! Y3(2) ! Y3(3) ! Y3(4) = X3 *
* |
|||||| Y ! Y (1) = Z ! Y (2) ! Y (3) = X *
* |
|||||| 2 2 2 2 2 2 *
* |
|||||| Y ! Y (1) = Z ! Y (2) = X *
* |
|||||| 1 1 1 1 1 *
* |
|||||| Y ! Y (1) = X *
* |
|||||||_0____0_______0______________________________________________________*
*________||________
Table 10.4
___________________________________________||
| B B B(1) B(2) . . . . . . . . .|
||.1;1 0;1 0;1 0;1 ||
||.. ||
| B B B(1) B(2) . . . A(k) |
||.1;1 0;1 0;1 0;1 1 ||
||.. ||
| B B B(1) B(2) A(3) |
| 1;1 0;1 0;1 0;1 1 |
| B B B(1) A(2) |
| 1;1 0;1 0;1 1 |
| (1) |
| B1;1 B0;1 A1 |
| |
||B1;1__A1_________________________________||
Table 10.5
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