Torsion in the Cohomology of Mapping Spaces
Mark W. Winstead
University of Virginia
April 30, 1993
Abstract
It is conjectured that under certain hypotheses, H*(Map (BV; X); ) is a free
Z=pm module when H*(X; Z=pm ) is a free Z=pm module. We will discuss
attempts by the author to prove this conjecture as well as other related result*
*s.
Contents
1 Introduction 1
2 Background 4
2.1 The Steenrod Algebra : : : : : : : : : : : : : : : : : : : : : : : : 4
2.1.1 Instability : : : : : : : : : : : : : : : : : : : : : : : : : : :*
* 5
2.1.2 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : 6
2.1.3 The Milnor Bocksteins : : : : : : : : : : : : : : : : : : : : 7
2.2 The Bockstein homomorphisms and BZ=p : : : : : : : : : : : : : 8
2.2.1 The Bockstein homomorphisms : : : : : : : : : : : : : : : 8
2.2.2 The Bockstein homorphisms on BZ=p : : : : : : : : : : : 8
2.3 The Bockstein spectral sequence : : : : : : : : : : : : : : : : : : 9
3 The T -functor and its structure 14
3.1 The T -functor : : : : : : : : : : : : : : : : : : : : : : : : : : : : :*
* 14
3.1.1 The existence of T : : : : : : : : : : : : : : : : : : : : : : 14
3.1.2 Properties of T : : : : : : : : : : : : : : : : : : : : : : : : 15
3.2 The internal structure of T : : : : : : : : : : : : : : : : : : : : : *
*17
3.2.1 Another useful definition : : : : : : : : : : : : : : : : : : 17
3.2.2 The action of Ap on T M : : : : : : : : : : : : : : : : : : 20
3.2.3 The operations on products : : : : : : : : : : : : : : : : : 22
3.2.4 The algebra structure of the operations : : : : : : : : : : 23
4 K"unneth relations 25
5 The integral cohomology 28
5.1 Bocksteins and H*(Map (B Z=p; X); Z) : : : : : : : : : : : : : : : 28
5.2 Plausibility: Application to Lie Groups : : : : : : : : : : : : : : : 31
6 Speculations 33
6.1 Connective Morava K-theory : : : : : : : : : : : : : : : : : : : : 33
6.2 The cohomology of Map (B Z=p; X)f : : : : : : : : : : : : : : : : 34
i
Chapter 1
Introduction
Since the mid 1980's, a popular topic in algebraic topology has been the study
and applications of the T-functors defined by Jean Lannes, which take a category
called U (orK) to itself. The history of these functors finds its roots in Hayn*
*es
Miller's work with the space of maps from an infinite dimensional space to
another space and his answer to the Sullivan conjecture. One real strength of
the T-functors lies in the property that there is a natural map
(1.1) TV H*(X; Z=p) ! H*(Map (BV; X); Z=p)
which is quite often an isomorphism. (By Map (X; Y ), we mean the space of
unbased maps from X to Y given the compact open topology.) It is this isomor-
phism which interests us, so Chapter 3 includes a discussion which summarizes
many of the known conditions for which the natural map is an isomorphism.
The author believes the following to be true:
Conjecture 5.1.1 Suppose X is a space such that the natural map (1.1) is an
isomorphism. If H*(X; Z=pm ) is a free Z=pm module, then H*(Map (BV; X); Z=pm )
is a free Z=pm module.
Let Zp denote the p-adic integers. If the previous conjecture is true, the foll*
*ow-
ing would be a corollary
Conjecture 5.1.5 Suppose that X is a space as in the previous theorem. If
H*(X; Zp) has no torsion, then H*(Map (BV; X); Z) also has no torsion.
This result would combine with a theorem of Dwyer and Zabrodsky to show
the next proposition. Clarence Wilkerson has informed me that he and Bill
Dwyer have observed this result to be true and from the context of his note it
appears that they used Borel's Theorem [Bo ].
Proposition 5.2.3 Suppose that G is a connected compact Lie group and that
g 2 G is an element of order p. Let ZG (g) be the centralizer of g in G. If
H*(BG; Z) has no p torsion, then neither does H*(BZG (g); Z), i.e. Conjecture
5.1.3 holds for BG.
CHAPTER 1. INTRODUCTION 2
The main result of this paper is the reduction of Conjecture 5.1.1 to a tech-
nical conjecture, which is mentioned stated later in this introduction. The
reduction utilizes the (mod p) Bockstein spectral sequence and the evaluation
map "X : BV x Map (BV; X) ! X. It is the evaluation map which induces
the natural map I have been referring to. The mathematics behind the case
m = 2 has proven to be the most interesting and may be of independent inter-
est to many readers. In order to show that case, I had to explore the internal
structure of TV . We will restrict our attention in this paper to V = Z=p and
set T = TV , since it can be shown quite easily that TW = T dimW.
Using a map from M to H*(BZ=p) T M which is analogous to the evalu-
ation map, one can define (Definition 3.2.1) a family of natural additive oper-
ations {ts}s2Z from a module M to T M. The operation ts sends an element
x 2 Mn to an element tsx 2 (T M)n-s. We will also give two alternate def-
initions for this family of operations, each of which has its advantages and
disadvantages. We now summarize many of the results of Chapter 3 with the
following theorem.
Theorem 1.0.1 Let M be an object of U.
a. If M has a presentation as an A-module with generators {xfl} and relations
{rfi}, then T M has a presentation with generators {tixfl} and relations
{tjrfi}.
b. The Steenrod algebra and the operations {ts} interact as follows:
P s-i k-i
(i)Suppose p = 2. Then tsSqk = i i Sq ts-i:
(ii)Suppose p > 2. Then
X [_s] - i(p - 1)
tsPk = 2 Pk-its-2i(p-1)
i i
and
t2rfi = fit2r+ t2r-1
t2r+1fi = fit2r+1:
(iii)Let p be any prime and let Qk be the kth Milnor Bockstein. Then
t2rQk = Qkt2r+ t2r-2pk+1
t2r+1Qk = Qkt2r+1
c. If {xfl}fl2 is a set of elements which generate M as a vector space, then
{tsxfl}fl;sgenerate T M as a vector space.
Parts a and c constitute Theorem 3.2.5, while part b is Theorem 3.2.7. Part
c can also be viewed as a corollary of parts a and b.
CHAPTER 1. INTRODUCTION 3
The case m = 2 in Conjecture 5.1.1 is the case k = 1 of the following
theorem. The cases k 2 help inspire Conjectures 6.1.1 and 6.1.2. The Milnor
Bocksteins {Q k} are important elements in the Steenrod algebra and they are
discussed in Chapter 2.
Theorem 3.2.8 If the kthMilnor Bockstein is zero on all of an unstable module
M, it is zero on all of T M.
As for the cases m > 2, we reduce Conjecture 5.1.1 to the following conjec-
ture.
Conjecture 5.1.3 For any space X, define
tr;s: H*(X; Z=ps) ! H*-r(Map (B Z=p; X); Z=ps)
by tr;s(a) = "*X(a)=hr;s, where = denotes the slant product and hr;sis non-zero
in Hr(B Z=p; Z=ps). Under the hypothesis on X in Conjecture 5.1.1, if the sth
Bockstein is zero on H*(X; Z=ps), then t2r-1;s 0 for all r.
Furthermore, we can show that this conjecture is true if there exist certain
operations, analogous to the Milnor Bocksteins, from mod Z=ps cohomology to
itself. The properties required of these operations are discussed in full in ch*
*apter
5.
We now outline the rest of this dissertation. Chapter 2 consists of a review*
* of
the prerequisite knowledge needed about the Steenrod algebra and the related
unstable categories, as well as the homology and cohomology of BZ=p and the
Bockstein spectral sequence. In chapter 3, the T-functor and its properties
are introduced, and we explore some of the internal structure of the T-functor.
Chapter 4 explores some K"unneth relations that we will need. In chapter 5, we
reduce Conjecture 5.1.1 to Conjecture 5.1.3 and examine some of its corollaries.
Chapter 6 discusses various conjectures and potential conjectures analogous to
Conjecture 5.1.1, including some involving the connective Morava K-theories, a
generalized cohomology theory useful in homotopy theory.
Chapter 2
Background
In this chapter, we will review much of background knowledge needed for the
rest of this paper. In section 1, we review the Steenrod algebra and some of
the related categories. In section 2, we recall the definition of the Bockstein
homomorphism and its effects on the cohomology of BZ=p. In section 3, we
discuss the Bockstein spectral sequence.
2.1 The Steenrod Algebra
Let A denote the mod p Steenrod algebra, the algebra of all natural stable
transformations from mod p cohomology to itself, or equivalently the mod p
cohomology of the mod p spectrum HZ=p. The algebra A is the quotient of the
free graded associative Fp-algebra with unit generated on elements:
o Sqiof degree i, if p = 2,
o fi of degree 1 and
o Piof degree 2i(p - 1), i > 0, if p an odd prime
by the ideal generated by the Adem relations
[i=2]Xj - k - 1
Sq iSqj- Sqi+j-kSqk
0 i - 2k
for all positive integers i and j such that i < 2j if p = 2 or fi2 = 0 and the
Adem relations
[i=p]X (p - 1)(j - k) - 1
PiPj - (-1)i+k Pi+j-kPk
0 i - pk
CHAPTER 2. BACKGROUND 5
for all positive integers i and j such that i < pj and
[i=p]X (p - 1)(j - k)
PifiP j- (-1)i+k fiP i+j-kPk
0 i - pk
[(i-1)=p]X (p - 1)(j - k) - 1
- (-1)i+k-1 Pi+j-kfiP k
0 i - pk - 1
for all positive integers i and j such that i pj if p is an odd prime.
The Steenrod algebra is a Hopf algebra, with the coproduct ffi : A ! A A
given by P
ffi(Sqk)= i+j=kSq i Sqjifp = 2;
ffi(fi)= fiP 1 + 1 fi;
and ffi(P k)= i+j=kP i Pjifp > 2;
where Sq0 and P0 denote the units of A for p = 2 and p is odd, respectively.
The coproduct allows us to define an A module structure on the tensor product
of two A modules M and N by utilizing the formula
0||m| 0
( 0)(m n) = (-1)| m n
for all ; 02 A, m 2 M and n 2 N.
2.1.1 Instability
The mod p cohomology of a space X has a certain property as a A module
which is called instability. More precisely, for X any space:
o Sqix = 0 if x 2 H*(X), i > |x| and p = 2;
o fifflPix = 0 if x 2 H*(X), ffl + 2i > |x|,
e = 0; 1 and p > 2,
where |x| denotes the degree of x.
Any A module M which satisfies this property is called unstable.
Note that any unstable module is trivial in negative degrees since we have
identified Sq0 (resp. P0) with the identity operation.
Unstable algebras
The mod p cohomology of a space X is naturally a graded communative algebra
over A satisfying two properties:
K1:
P j i-j
Sqi(xy) = P j(Sq x)(Sq y) forx; y 2 H*(X) if p = 2,
Pi(xy) = j(P jx)(P i-jy);
fi(xy) = (fix)y + (-1)|x|xfiy forx; y 2 H*(X) if p > 2.
CHAPTER 2. BACKGROUND 6
The property (K1) is known as the "Cartan formula".
K2:
Sq|x|x = x2 for anyx 2 H*(X) ifp = 2;
P|x|=2x = xp for any x with even degreeHin*(X) ifp > 2:
This leads to the following definition.
Definition 2.1.1Suppose an unstable module K has maps : K K ! K
and j : Fp ! K which give K a communative, unital, Fp-algebra structure. K
is then called an unstable algebra if properties K1 and K2 hold.
The Unstable Categories
Denote the category whose objects are graded unstable Ap-modules and whose
morphisms are Ap-linear maps of degree zero Up, or, if p is understood, U and
let Kp, or simply K, denote the category whose objects are unstable algebras
and whose morphisms are Ap-linear algebra homomorphisms. We concentrate
on the category U.
Two main properties of U are that it is abelian and that it has enough
projectives. This latter property is implied by
Proposition 2.1.2[SE ] [MP ] There is an unique (up to isomorphism) unstable
module F (n) with a class in in degree n such that the natural transformation
from Hom U(F (n); M) to Mn defined by f 7! f(in) is an equivalence of functors.
The functor M 7! Mn is right exact, so F (n) is projective. F (n) is often
called the free unstable module on a generator of degree n.
2.1.2 Examples
Preliminary Remark
As already mentioned, the mod p cohomology of a space X is an example of
a module over the mod p Steenrod algebra. Before continuing with specific
examples, we now remark about the nature of Sq1 for the case of p = 2 and fi
for the case p > 2. On the mod p cohomology of a space, this operation is the
same as the Bockstein homomorphism induced by the short exact sequence
0 ! Z=p ! Z=p2 ! Z=p ! 0:
(Recall that any short exact sequence of groups induces a long exact sequence
in both homology and cohomology.)
CHAPTER 2. BACKGROUND 7
BZ/p
The mod 2 cohomology of BZ=2 is one example of an unstable algebra of partic-
ular interest in the study of the Steenrod algebra and this paper. H*(B Z=2) is
the polynomial algebra F2[x], where x is the generator in degree 1. The action
of A is completely determined by properties K1 and K2.
The mod p cohomology of B Z=p is also an unstable algebra of interest.
H*(B Z=p) is the tensor product of an exterior algebra E(u) with generator u
of degree 1 and a polynomial algebra Fp[v] with generator v of degree 2. The
action is determined by the two properties K1 and K2 and by the fact that fi is
the Bockstein homomorphism, i.e. fiu = v.
The importance of these examples in this paper will become clear. Their
importance in the study of the Steenrod algebra deserves some comment now.
o Let us define P (n) to be the product of n copies of BZ=2 and let w be the
product x1: :x:n, where xiis the degree 1 generator of the cohomology of
the ith copy of BZ=2 in P (n). The map A2 ! H*(P (n); Z=2) given by
7! w is a monomorphism in degrees less than or equal to n. There is a
similar statement for odd primes.
o Milnor [Mil] uses the cohomology of BZ=p and its finite nth skeletons to
determine the structure of the dual of Ap, which he then uses to study
Ap.
In chapter 3, we will need to know the homology structure of BZ=p. B Z=p
is a Hopf space, a topological space with a group structure, and this group
structure induces a product on the homology of BZ=p. For p = 2, the mod
2 homology of B Z=2 is a divided power algebra: an algebra with additive
generators {ai} indexed by a subset of the non-negative integers and prod-
uct given by aiaj = i+jiai+j. In particular, it is a divided power alge-
bra on the non-zero elements hr 2 Hr(B Z=2; Z=2). For odd primes, let u 2
H1(B Z=p; Z=p) and v 2 H2(B Z=p; Z=p) be as before, when we described the
mod p cohomology of BZ=p. Then the mod p homology of BZ=p is the ten-
sor product of an exterior algebra generated by an element h1 and of a di-
vided power algebra generated by elements h2; h4; h6; : :;:h2i; : :,:where h1 2
H1(B Z=p; Z=p) and h2k 2 H2k(B Z=p; Z=p) are such that < h1; u >= 1 2 Z=p
and < h2k ; vk > = 1 2 Z=p.
2.1.3 The Milnor Bocksteins
In the paper that we just referenced, Milnor introduces the elements Qi 2 A2pi-*
*1,
which we shall use heavily in this paper (Theorem 3.2.8, et al). We now recall
their definition and some of their properties.
Definition 2.1.3Forip = 2 (p >i2), set Q0 = Sq1 (fi) and define Qi by
Qi= [Qi-1; Sq2] (Qi= [Qi-1; Pp]), where [; 0] = 0- (-1)|||0|0
CHAPTER 2. BACKGROUND 8
Among the properties of the Milnor Bocksteins are:
o They commute with one other and their square is trivial, i.e. the subal-
gebra of A generated by them is an exterior algebra.
o They are derivations, i.e. Q i(xy) = Qi(x)y + xQ i(y):
k i+1
o If x 2 H*(B Z=2; Z=2), then Qix = x2i+1and therefore Qixk = 1xk+2 -1.
Similarly, if ufflvk 2 H2k+ffl(B Z=p; Z=p), then
ae
0 ifffl = 0
(2.1) Q iufflvk = vk+pi ifffl = 1
P
o AQ iA = ji AQ j.
For proofs and more details on the Steenrod algebra and the categories Up
and Kp, see [McC ] [Mil] [MT ] [SE ] [Sw ] and [Sch].
2.2 The Bockstein homomorphisms and B Z=p
2.2.1 The Bockstein homomorphisms
Proposition 2.2.1For any short exact sequence of abelian groups,
E: 0 ! A i!B p!C ! 0;
and any space X, there are long exact sequences
. .H.n+1(X; C) d!Hn(X; A) i*!Hn(X; B) p*!Hn(X; C) d!Hn-1(X; A) . . .
and
. .H.n-1(X; C) d!Hn(X; A) i*!Hn(X; B) p!Hn(X; C) d!Hn+1(X; A) . .:.
The homomorphism d is commonly called the Bockstein homomorphism asso-
ciated to the given short exact sequence.
2.2.2 The Bockstein homorphisms on B Z=p
In this paper, we will need to understand various Bockstein homomorphisms on
the cohomology of BZ=p. A necessary first step is to know the cohomology of
BZ=p with various coefficients. The integral homology or cohomology of BZ=p
can be found either by computing it or by referring to sources such as [Wh ]. In
either case, we find that
8
< Z n = 0
(2.2) Hn(B Z=p) ~=: Z=p n > 0 and odd
0 otherwise
CHAPTER 2. BACKGROUND 9
and 8
< Z n = 0
(2.3) Hn(B Z=p) ~=: Z=p n > 0 and even
0 otherwise.
An application of the universal coefficient theorem gives us:
8
< Z=pi n = 0
(2.4) Hn(B Z=p; Z=pi) ~=: Z=p n > 0
0 otherwise
8
< Z=pi n = 0
(2.5) Hn(B Z=p; Z=pi) ~=: Z=p n > 0
0 otherwise.
It is an easy exercise using (2.4), (2.5) and induction on i to show:
Proposition 2.2.2For the short exact sequence
0 ! Z=pj ! Z=pk ! Z=pk-j ! 0
and the space BZ=p, the Bockstein homomorphism for homology
d : Hi(B Z=p; Z=pk-j) ! Hi-1(B Z=p; Z=pj)
is a group isomorphism for i even and the zero homomorphism for i odd, while
the Bockstein homomorphism for cohomology
d : Hi(B Z=p; Z=pk-j) ! Hi+1(B Z=p; Z=pj)
is a group isomorphism for i odd and the zero homomorphism for i even.
2.3 The Bockstein spectral sequence
Let X be any space. The mod p Bockstein spectral sequence (BSS) on X is a
natural singly graded spectral sequence perhaps best defined by using the exact
couple :
H*(X; Z) !i* H*(X; Z)
fi - . ae*
H*(X; Z=p):
induced from the exact sequence 0 ! Z !i Z !aeZ=p ! 0, where fi is the
associated Bockstein. The BSS converges to (H*(X; Z)=T orsion)Z=p: Unlike
most other spectral sequences, this ones indetermediate steps tell us something
useful: A cycle which survives to Ernand is contained in the image of dr comes
from a direct summand Z=pr of Hn(X; Z). For more on exact couples and
CHAPTER 2. BACKGROUND 10
spectral sequences, see [McC ],[MT ] or [Wh ]; for more on the BSS, see [B ] or
[MT ].
There are, however, technical difficulties in using this form of the BSS for
our purposes, which we will not elaborate on. In [MT ], an alternate procedure
is given. While for simplicity [MT ] only discusses it for p = 2, a careful che*
*ck
shows the analogous result for odd primes also holds. It was this version that
we used. A near equivalent to this version arises as follows, where we look at *
*the
spectral sequence in terms of spectra. Recall that an element in H*(X; R) can
be thought of as a map, unique up to homotopy, from the suspension spectrum
of X to the Eilenberg-MacLane spectrum with coefficients R, HR.
The short exact sequence 0 ! Z xp!Z ! Z=p ! 0 induces a fibration
of spectra HZ xp! HZ ! HZ=p, which provides us with the exact couple
mentioned above and with a long cofibration/fibration sequence (either since
we are discussing spectra):
: :j:1!-1HZ=p k1!HZ i1!HZ j1!HZ=p k1!HZ : : :
where i1 is alternate notation for the mapPxp, j1 is the "reduction" to HZ=p,P
k1 is the map to the cofiber of j1 and is the suspension functor (and -1 the
"desuspension" functor). The first differential d1 is the composite j1k1. Therer
are similar fibration sequences arising from short exact sequences 0 ! Z xp!
Z ! Z=pr ! 0 which also induce (co)fibration sequences
: :j:r!-1HZ=p kr!HZ ir!HZ jr!HZ=pr kr!HZ : :;:
where the various maps are defined analogously to the case r = 1 discussed
above. At each stage of the spectral sequence, an element lifts if the differen*
*tial is
zero modulo some indetermancy. The mthdifferential of the Bockstein spectral
sequence, defined on lifts of elements from the case r = 1, is dm = jm km . One
sees the liftings arise by chasing the following commutative diagram of maps.
r jr
: :!: HZ xp! HZ ! HZ=pr ! : : :
" xp k " aer
r+1 jr+1
: :!: HZ xp! HZ ! HZ=pr+1 ! : : :
In what follows below, we will define the differentials in our viewpoint of the
Bockstein spectral sequence by fir = ae1ae2: :a:er-1dr. The advantage of this
viewpoint is that fir can be seen as the Bockstein homomorphism arising from
the short exact sequence
0 ! Z=p ! Z=pr+1 ! Z=pr ! 0:
All of what follows can be shown by diagram chasing. The diagram on the
last page of this chapter has been provided for the reader's convienence.
CHAPTER 2. BACKGROUND 11
Summary of how the Bockstein spectral sequence works from this
perspective: For any space X, let ^a2 Hn(X; Z=pi) be an element which maps
to an element a 2 Hn(X; Z=p) under the reduction map Z=pi! Z=p. If fii^a= 0
modulo indetermancy, then there is an element of Hn(X; Z=pi+1) which hits ^a
under the homomorphism aei and a under the composite. We can refer to such
an element by calling it a lift of a to the i + 1 step. Continuing in this way,*
* one
can see that for any n, if an element a 2 Hn(X; Z=p) has a lift for all i, then
it is either in the image of fik for some k or a is a non-zero permanent cycle,
i.e. it is the image of an additive generator of an integral direct summand of
Hn(X; Z). If at the ith stage, fii^ais not zero modulo indetermancy, then a is
the image of an additive generator of a Z=pi direct summand of Hn(X; Z).
We end this chapter with some special cases and consequences of the Bock-
stein spectral sequences.
Theorem 2.3.1 Let X be a space whose integral cohomology is of finite type
(in each dimension, the cohomology is finitely generated).
a. H*(X; Z) has no p-torsion if and only if H*(X; Z=pm ) is a free Z=pm -
module for all m 1.
b. The following are equivalent:
i) H*(X; Z=pm ) is a free Z=pm -module.
ii)fi1; : :;:fim-1 are identically zero on the lifts of elements from H*(*
*X; Z=p).
iii)The reduction map ae : Z=pm ! Z=p induces an epimorphism from
H*(X; Z=pm ) to H*(X; Z=p).
Furthermore, if any of the above hold, the free Z=pm module H*(X; Z=pm )
is additively generated by lifts of elements of H*(X; Z=p).
Proof: Part a and the case i ( ii of part b is a diagram chase. The case i ,
iii is a diagram chase using the universal coefficient theorem and the definiti*
*on
of Tor. Recall that the universal coefficient theorem says that
0 ! Hn(X; Z) R ! Hn(X; R) ! T or(Hn+1(X; Z); R) ! 0;
while to define Tor(R; R0) one takes an exact sequence
0 ! K f!F !g R ! 0
where K and F are free and tensoring the sequence with R0. Tor(R; R0) is then
defined as the kernel of f 1R0. The case i ) ii of part b can also be done as
a diagram chase or as follows: If H*(X; Z=pm ) is a free Z=pm module, the uni-
versal coefficient theorem implies that H*(X; Z=pm-l) is a free Z=pm-l module
CHAPTER 2. BACKGROUND 12
for all l such that 0 l m - 1. A diagram chase of either the Bockstein spec-
tral sequence diagram or a diagram involving the universal coefficient theorem
applied twice yields that the reduction map ae : Z=pm ! Z=pm-1 ! 0 induces
H*(X; Z=pm ) ae*!H*(X; Z=pm-1 ) ! 0
under the hypothesis. The reduction map fits inside the short exact sequence
that induces fim-1 , hence fim-1 has all of H*(X; Z=pm-1 ) as its kernel. Thus
there is an induction argument to show the case i ) ii of part b.
The question arises that this is modulo indetermancy. The stronger state-
ment made is accurate. The indetermancy for
fim-1 : H*(X; Z=pm-1 ) ! H*(X; Z=p)
comes the map from H*(X; Z=pm-2 ) to H*(X; Z=pm-1 ) induced by the inclu-
sion of Z=pm-2 into Z=pm-1 . By naturality, the composite of the inclusion and
fim-1 , which sends H*(X; Z=pm-2 ) to H*(X; Z=p) must be either fim-2 or the
zero map, but fim-2 is zero, by induction. Actually, it can be shown that the
composite must be fim-2 , see [MT ]. Thus i , ii.
As for the claim about lifts additively generating H*(X; Z=pm ), we have the
long exact sequence of cohomology induced by
0 ! Z=pm-1 ! Z=pm ! Z=p ! 0;
which tells us that H*(X; Z=pm ) is generated by elements which either are
the image of elements from H*(X; Z=pm-1 ) or map to non-zero elements of
H*(X; Z=p). However, no direct summand of H*(X; Z=pm ) is generated by
an element in the image of H*(X; Z=pm-1 ), since H*(X; Z=pm-1 ) and its im-
age are annihilated by pm-1 and H*(X; Z=pm ) is a free Z=pm module. Hence
H*(X; Z=pm ) is generated by lifts. 2
CHAPTER 2. BACKGROUND 13
This page blank !!!!!!
Chapter 3
The T -functor and its
structure
In this chapter, we introduce Lannes' functor T and study the action of the
mod p Steenrod algebra Ap on the image of M under the functor T , T M. We
will see that knowledge of the Ap-module structure of M has a lot to say about
the Ap-module structure of T M.
3.1 The T -functor
Let V be an elementary abelian p-group, or a finite dimensional vector space
over Fp. The functor TV is the left adjoint to tensoring with H*(BV ) = H*(V )
in the category Up, i.e. TV is that unique functor such that for any two objects
M and N in Up, Hom Up(TV M; N) ~= Hom Up(M; H*(V ) N). There is a
similarly defined left adjoint0TV0in the category Kp, and if M 2 Kp, then Lannes
has shown that as modules TV M ~=TV M.
3.1.1 The existence of T
Jean Lannes cites in [L3] three different approaches to showing the existence
of TV . Each approach has its advantages and disadvantages in showing various
results and properties. One method which is of particular interest in this paper
is that of J. F. Adams [Ad3 ]. We now describe it.
Let M denote the category whose objects are (stable) modules over Ap and
whose morphisms are the Ap-linear maps of degree zero. Note that U is a full
subcategory of M. Suppose that M and N are objects of M and furthermore
suppose that Nn = 0 for n less than some integer n0. Let H* = H*(V ) =
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 15
H*(BV ). There is an isomorphism
Hom M (M; H* N) ~=Hom M (M H*; N);
where H* denotes the dual of H*, i.e. H* ~= H*(BV ) (viewed as being in
negative degrees) with Ap-module structure on H* given by
: t = t: O fort 2 H* and 2 Ap
where O : Ap ! Ap denotes the canonical anti-isomorphism.
The forgetful functor 1 : U ! M has a left adjoint denoted by 1 and
referred to by Lannes in [L3] as the destablization functor. Explicitly, if M i*
*s an
object of M, then 1 M is the universal unstable quotient of M. For example,
1 (nAp) is F (n). The adjunction then implies that we can define TV M as
1 (M H*).
To close this section, we point out that since 1 is the left adjoint of the
forgetful functor from the category of unstable modules to the category of all
modules over Ap, we can observe that 1 M is the universal unstable quotient
of M, i.e. the largest quotient of M which is unstable.
3.1.2 Properties of T
In all that follows, we will use T to mean TZ=p.
Theorem 3.1.1 The functor TV has the following properties [L1], [L2], [L3]:
(a) TV ~=T dimV.
(b) T is exact. (It is right exact since it is a left adjoint.) Thus T takes d*
*irect
sums to direct sums.
(c) T commutes with colimits.
(d) T commutes with tensor products.
(e) Suppose that M is a locally finite module, i.e. if for all m 2 M, the
submodule generated by m is finite. Then T M ~=M.
We will not prove this theorem here, one reason being that to repeat a proof
here would roughly double the length of this paper. However, do note that some
of it follows from the fact that T is a left adjoint. Part (a) automatically gi*
*ves
us some extensions of the results included herein which we do not specifically
mention. Throughout the sequel, we restrict our attention to T .
One property of T of importance in this chapter is contained in the following
proposition.
PropositionL3.1.2For any n, the module T (F (n)) is isomorphic to the direct
sum ni=0F (i).
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 16
Proof: Let H* = H*(B Z=p; Z=p). Then for all unstable modules M,
(3.1) Hom Up(T F (n); M)~= Hom Up(F (n); H* M)
~= (H* M)n
Mn
~= Mn-r
r=0
Mn
~= Hom Up(F (n - r); M)
r=0
Mn
~= Hom Up( F (n - r); M):
r=0
L n
Alternately setting M equalLto T F (n) and r=0F (n - r), we see by standard
arguments that T F (n) and nr=0F (n - r) are isomorphic. 2
One additional property mentioned earlier in this paper is now repeated here
in greater detail. Consider the evaluation map
"X : BZ=p x Map (B Z=p; X) ! X
defined by taking the point (x; f) to f(x). This map induces a homomorphism
from H*(X; Z=p) to H*(B Z=p; Z=p) H*(Map (B Z=p; X); Z=p), which is an el-
ement of Hom Up(H*(X; Z=p); H*(B Z=p; Z=p) H*(Map (B Z=p; X); Z=p)) and
gives us an element j in Hom Up(T H*(X; Z=p); H*(Map (B Z=p; X); Z=p)). Un-
der many different sets of conditions, j, which is the natural map mentioned
throughout this paper, is an isomorphism. It is this property which many con-
sider the most important, as it allows one to study H*(Map (B Z=p; X); Z=p)
in a new way. We now include a discussion of many of these sets of condi-
tions. Throughout this discussion, assume that all cohomology, unless otherwise
stated, is mod p cohomology.
Let ^Xdenote the Bousfield-Kan p-completion of X [BK ].
Theorem 3.1.3 [L3] Let X be a space such that both the unstable Ap-algebra
H*(X) and TV H*(X) are of finite type and TV H*(X) is trivial in degree 1.
Then the natural map
TV H*(X) ! H*(Map (BV; ^X))
is an isomorphism.
Theorem 3.1.4 [DZ ] Let X be a space such that the unstable Ap-algebra H*(X)
is of finite type. If ss1X is a finite p-group, then the canonical map
H*(Map (BV; ^X)) ! H*(Map (BV; X))
is an isomorphism.
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 17
The nature of the maps in these two theorems give us one set of sufficient
conditions, i.e. the composite of the maps in these two theorems give us the
natural map j : TV H*(X) ! H*(Map (B Z=p; X)) defined previously.
Another set of conditions for which j is an isomorphism is given to us by
the work of Dror-Farjoun and Smith [DS ]
Theorem 3.1.5 Suppose that X is a connected nilpotent p-local space which
has only finitely many non-trivial homotopy groups. If ssn(X) is finite for all*
* n,
then j is an isomorphism.
There is a third set of conditions, which we have chosen to mention last,
because we wished to state more concrete conditions first.
Theorem 3.1.6 [L3] Suppose X and Y are two spaces, each with mod p coho-
mology of finite type, and suppose there is a map ! : BV x Y ! X. Then the
following two conditions are equivalent:
a. the unstable Ap-algebra homomorphism TV H*(X) ! H*(Y ), adjoint to
!* : H*(X) ! H*(BV ) H*(Y ), is an isomorphism;
b. the map ^Y! Map (BV; ^X) induced by ! is a homotopy equivalence.
This almost says that if a space smells and behaves like Map (BV; X), then
it is Map (BV; X).
3.2 The internal structure of T
To study the structure of the module T M when we understand the structure
of M, we need tools to examine the individual elements of T M. We choose
to present Adams' proof of existence rather than referring the reader to other
proofs because Adams' approach has the advantage that we can view individual
elements quite easily. Let hr be an additive generator of Hr(B Z=p; Z=p) and
let m denote an element of M. Then hr m 2 H*(B Z=p; Z=p) M projects to
an element [hr m]2 1 (H*(B Z=p; Z=p) M) : We will often abuse notation
and simply write hr m when we mean the class [hrP m]. In [Ad3 ], Adams
gives necessary and sufficient conditions for rhr mr 2 H*(B Z=p; Z=p) M
to map to zero in 1 (H*(B Z=p; Z=p) M)
3.2.1 Another useful definition
Let ff be defined as that map in Hom Up(M; H* T M) which is the image of
the identity from T M to T M under the adjunction isomorphism
Hom Up(M; H* T M) ~= Hom Up(T M; T M).
Definition 3.2.1P(a) Let p = 2. For m 2 Mn and r any integer, define trm
by ff(m) = r(xr trm). Note that trm is well-defined.
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 18
(b) Let p be an odd prime. For m 2 Mn , r any integer, ffl = 0 or1, u an
additivePgenerator of H1(Z=p) and v = fiu 2 H2(Z=p), define t2r+fflm by
ff(m) = r;ffl=0;1(ufflvr t2r+fflm). Note that tsm depends on the choice*
* of
u, but as u is unique up to multiplication by elements of Z=p* = Z=p-{0},
so is tsm.
Remarks
(a) For p = 2 (p > 2), if r < 0 or if r > |m| (r > [|m|=2]), trm = 0
(t2r+fflm = 0)
(b) Throughout the rest of the paper, we will assume that t2r+fflis well-defin*
*ed
for odd primes.
The following theorem holds.
Theorem 3.2.2 trm = [hr m], where hr is chosen as in chapter 2.
Before beginning the proof, we make the following observation.
Lemma 3.2.3 Let U; V be two finite dimensional vector spaces over Fp and let
V *be the dual of V . Furthermore, let the set S V be a basis for V . The
map OE 2 Hom Fp(U; V * V U) adjoint to the identity on U V is given
by X
OE(u) = v* v u:
v2S
Proof: The isomorphismP : Hom Fp(V U; V U) ! Hom Fp(U; V * V U)
is defined by (f)(u) = v2Sv* f(v u) 2
Proof of the theorem: The theorem follows from the lemma, the definition
of trm , the fact that H*(B Z=p; Z=p) is of finite type and Adam's definition of
T M. Specifically, we said that trm is given by
X
m 7! ufflvk t2k+fflm
while T M = 1 (H*(BZ=p; Z=p) M). 2
Before continuing, let us return to Proposition 3.1.2. For the next theorem
which shows the usefulness of our definitions, we need the following lemma.
L n
Lemma 3.2.4 There is an isomorphism r=0F (n - r) ! T F (n) which sends
n-r to trn
Proof: Recall the isomorphisms given by 3.1 in the proof of Proposition 3.1.2.
Let M = T F (n) and let us chase through through the isomorphisms, start-
ing with the identity map on T F (n). By our definition of tr, the identity
mapPis sent to the map in Hom Up(F (n); H* T F (n)) determined by n 7!
ufflvk t2k+ffl. Under the next isomorphism, this map is sent to the element
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 19
P
ufflvk t2k+ffl2 (H*L T F (n))n. The following isomorphism sends this
element to the map in rHom Up(F (n-r); T F (n)) determined by n-r 7! trn.
The lemma then follows. 2
Theorem 3.2.5 a) If M is an object of Up with {mfl} as a set of homogenous
generators of M and {rfi} as a set of relations on M such that the sets
describe a presentation of M, then T M has a presentation with generators
{timfl} and relations {tjrfi}.
b) If the set {mfl} additively generates an unstable module M as a vector
space, then the set {tsmfl} additively generates T M as a vector space.
Proof: Part a: Note that Lemma 3.2.4 shows part a for M = F (n), which has
a single generator n and no relations (except for those forced by unstability),
since T F (n) has generators {tin} and no relations.
The theorem then follows for all M since specifying sets of generators and
relations is equivalent to defining a right exact sequence
M rfiM mfl
F (nfi) ! F (nfl) ! M ! 0:
fi fl
Since T is right exact and commutes with direct sums, we have
M M
T F (nfi) ! T F (nff) ! T M ! 0:
fi ff
Part b: By using the identification of hr m with trm, we see that an
element of T M can be written as a sum of elements of the form tkmk, so it
suffices to show that an element trm can be written as a sum of elements from
the set {tsmfl}. Since m is a sum of elements from the set {mfl} by hypothesis,
part b follows. 2
The proof of part a of theLprevious theorem suggests a third way to define
trm. Recall that T F (n) ~= nk=0F (k). Define jk to be the inclusion of F (k)
into T F (n). We noted in Chapter 2 that an element m 2 Mn is the image of
n under some unique map f : F (n) ! M. Define the element orm 2 (T M)n-r
by the equation orm = (T f) O jn-rn.
Lemma 3.2.6 orm = trm.
Proof: We can see from the proof of part a of Theorem 3.2.5 that it suffices
to show the theorem for M = F (n). It should be clear that by definition
orn = n-r, and in Theorem 3.2.5, we saw that trn = n-r. The lemma
follows. 2
There is yet another way to define trm if for some space X, M = H*(X; Z=p)
and the natural map from T H*(X; Z=p) to H*(Map (B Z=p; X); Z=p) is an iso-
morphism. For the moment, let X and Y be any two spaces. There exists an
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 20
external product from H*(Y ; R) H*(Y x X; R) to H*(X; R) called the slant
product, which takes an element (a; b) 2 Hq(Y ; R) Hp(Y x X; R) to an ele-
ment b=a 2 Hp-q(X; R). The slant product is described in detail in many basic
homology and cohomology texts.
Now let X be our X, Y be BZ=p and G be Z=p. Define tr;1m to be the ele-
ment ffl*X(m)=hr, where fflX is the evaluation map described in section 3.1. Re*
*call-
ing that ffl*Xinduces the natural map j from T H*(X; Z=p) to H*(Map (B Z=p; X);*
* Z=p),
we see that tr;1m = trm when the natural map is an isomorphism, i.e. we have
X X
hr (ufflvk t2k+fflm) 7! < hr; ufflvk > t2k+fflm = trm:
Remark In a later chapter, we will generalize the notion of tr;1.
3.2.2 The action of Ap on T M
The next theorem determines the action of Ap on T M in terms of the action
on M.
P r-i k-i
Theorem 3.2.7 (a) Suppose p = 2. Then trSqk = i i Sq tr-i:
(b) Suppose p > 2. Then
X [s=2] - i(p - 1)
tsPk = Pk-its-2i(p-1)
i i
and
t2rfi = fit2r+ t2r-1
t2r+1fi = fit2r+1:
(c) Suppose p is any prime. Then
t2rQk = Q kt2r+ t2r-2pk+1
t2r+1Qk = Q kt2r+1:
Proof: Let m 2 M, M any object of Up. Since ff is an Ap-module homomor-
phism, we have ff(m) = ff(m) for all 2 Ap. The theorem is basically a
consequence of this relation. Assume p = 2. On the one hand we have
P
Sqkff(m) = SqkPP rxr trm
= P r P iSqixr Sqk-itrm
= r i rixr+i Sqk-itrm
P k
while on the other we have ff(Sqkm) = rxr trSq m: If we compare the two
equations term by term, part (a) follows. The proof of part (b) is almost the
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 21
same. To prove part (c), let us again simplify by assuming p = 2. Recalling
that Qk is a derivation, we have
P
Q kff(m) = QkP rxr trm
= P r(Q kxr trm + xr Qktrm
= r( r1xr+2k+1-1 trm + xr Qktrm)
P
while ff(Q km) = rxr trQ km. Part(c) follows for p = 2 and a similar proof
works for odd primes. 2
Remark It is worth noting that the previous theorem combined with part a of
Theorem 3.2.5 yields a proof of part b of Theorem 3.2.5 which does not resort to
the identification of trm and [hrm]. For simplicity, let us restrict our attent*
*ion
to the case p = 2. The proof for odd primes is very similar. By Theorem 3.2.5a,
the set {tsxfl} will generate T M as a module over the Steenrod algebra. Hence
it shall suffice to verify part b on the elements of the form Sqktrx, where x is
one of the additive generators of M. We use induction on r. By Theorem 3.2.7,
X r - i
Sqktrx = trSqkx + Sqk-itr-ix:
i>0 i
For r = 0, one has Sqkt0x = t0Sqkx, and Sqkx can be written in terms of the
specified additive generators of M. So the result follows by the additivity of *
*t0.
If the result has been shown for values less than r, then the right summand of
the equation above can be written in terms of the set {tsxfl}. All that is left*
* to
do is to write trSqkx in term of the set. Again Sqkx can be written in terms
of the specified additive generators of M. The additivity of tr lets us complete
the induction argument.
The next theorem is in some ways the main theorem of this chapter in the
context of this dissertation and questions that arise from it motivate much of
the rest of this paper.
Theorem 3.2.8 Suppose M is an object of Up with the property that the nth
Milnor Bockstein Q nis zero on Mi for all i j for some j. Then Q nis zero
on (T M)i for all i j.
Proof: By Theorem 3.2.7, we can see that our hypothesis implies that
Qnt2r-1m = 0 and Q nt2rm = -t2r-2pn+1m for all appropriate r and all
m 2 M such that |m| j. We will further show that our hypothesis implies
that t2l-1m = 0 for all m 2 M of high enough dimension and all l 2 N . Then
we will have shown that Qnt2rm = -t2r-2pn+1m = 0, completing the proof.
Observe that the definition of the Milnor Bocksteins implies that if Qn is z*
*ero
on elements of dimension larger than j in an Up-module M, then Qk is zero on
such elements of M for all k n. Thus for all k n, Qkt2rm = -t2r-2pk+1m,
which implies that Qkt2r+2pk-2m = -t2r-1m. The dimension of t2r+2pk-2m is
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 22
negative (|m| - 2r - 2pk + 2) for large enough k, hence t2r+2pk-2m = 0 and if
k n, -t2r-1m = Q kt2r+2pk-2m = 0. By choosing k max(n; logp((|m| - 2r + 2)=2)*
* + 1 ),
we are done. 2
Corollary 3.2.9If the natural map j from T H*(X; Z=p) to H*(Map (B Z=p; X); Z=p)
is an isomorphism and H*(X; Z) has no Z=p direct summands, i.e. fi (Sq1 if
p = 2) is zero on H*(X; Z=p), then H*(Map (BZ=p; X); Z) has no Z=p direct
summands.
Proof: It follows from the fact that fi is the first differential of the Bocks*
*tein
spectral sequence (see Chapter 2). 2
Remarks
(a) Theorem 3.2.8 is a generalization of the following unpublished observation
of Nick Kuhn:
Observation 3.2.10 (Kuhn) Suppose M is an object in U2 and suppose
that Sq1 0 on M. Then Sq1 0 on TM.
This case of Theorem 3.2.8 can be shown by observing that the doubling
functor and the suspension functor both commute with the T functor.
(b) It is worth noting that the following lemma, which is cited in [Sch], is v*
*ery
similar to Theorem 3.2.8 in style of proof. I came across this lemma only
days after proving Theorem 3.2.8.
Lemma 3.2.11 Let p be an odd prime and T 0(-), the left adjoint to
H*(CP 1 ; Z=p) (-) in U0, the subcategory of Up whose objects are con-
centrated in even dimensions. If M is in U0, then T M ~=T 0M in Up.
3.2.3 The operations on products
Recall that Kp is the full subcategory of Up whose objects are the unstable Ap
algebras . Then for M 2 Kp, ff : M ! H* T M is an algebra homomorphism,
hence ff(mn) = ff(m)ff(n). For p = 2, this is the same as
X X X
xr tr(mn) = ( xi tim)( xj tjn);
which implies that X
trmn = (tim)(tjn):
i+j=r
If p > 2, we have
X X 0 X 00
ufflvr t2r+ffl(mn) = ( ufflvi t2i+ffl0m)( ufflvj t2j+ffl00n);
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 23
which yields
X
t2rmn = (t2im)(t2jn); (recall thatu2 = 0)
i+j=r
and X
t2r+1mn = {(t2i+1m)(t2jn) + (-1)|m|(t2im)(t2j+1n)}:
i+j=r
Summarizing:
Theorem 3.2.12 Let M be an object of Kp. Then
P
(a) If p = 2, trmn = i+j=r(tim)(tjn):
(b) If p > 2, X
t2rmn = (t2im)(t2jn)
i+j=r
and
X
t2r+1mn = {(t2i+1m)(t2jn) + (-1)|m|(t2im)(t2j+1n)}:
i+j=r
3.2.4 The algebra structure of the operations
In this section, we examine some of the additional structure of the ti's. In
particular, let V and W be any two vector spaces. For any homomorphism j
from V to W , there exists a natural transformation j! from the functor TV to
the functor TW . This functor is defined as follows: j induces a homomorphism
from H*(W ) to H*(V ), which in turn induces maps
Hom Up(M; H*(W ) N) ~= Hom Up(TW M; N)
# #
Hom Up(M; H*(V ) N) ~= Hom Up(TV M; N):
Hence there is a natural map induced by j from TV M to TW M for all M. Using
Adams' proof of the existence of TV M, we can give this map more precisely. Our
map j induces a homomorphism j* from H*(BV ) to H*(BW ), which in turn
gives a map j*1M from H*(BV )M to H*(BW )M. Thus understanding the
map j* allows us to understand the map TV M ! TW M induced by j : V ! W
Let us examine what happens when we restrict our attention to the case
where V = Z=pxZ=p, W = Z=p and the homomorphism j is addition. Recalling
that TV ~= T 2, we see a way of composing two t* operations. The resulting
algebra then can be studied. More specifically, the addition homomorphism
induces a homomorphism from H*(B Z=p) H*(B Z=p) to H*(B Z=p) given by
CHAPTER 3. THE T -FUNCTOR AND ITS STRUCTURE 24
i+j
hi hj 7! i hi+j. This allows us to think of the t* operations as operations
from T M to T M since
j!tr(tsm) = j![hr [hs m]] = [j*(hr hs) m]:
Thus we could redefine, if we choose to do so, the operations t* as operations
from T - to T -, and the "old" tr would become the inclusion of the module
into T -, followed by the "new" tr.
Recall the definition of a divided power algebra: it is an algebra with addi*
*tive
generators {ai} indexed by a subset of the non-negative integers and product
given by aiaj = i+jiai+j.
Theorem 3.2.13 a) When p = 2, the t* operations form a divided power al-
gebra over Z=2. The algebra is generated by the operations t2ifor integers
i 0 with t2it2i= 0. More precisely, titj = i+jiti+j.
b) When p is an odd prime, the t* operations with even index form a divided
power algebra over Z=p. The algebra of all tiis generated by the operations
t1 with t1t1 = 0 and the operations t2pifor integers i 0 and (t2pi)p = 0.
More precisely,
i+j
(i)t2it2j= i t2(i+j)
i+j
(ii)t2it2j+1= i t2(i+j)+1
i+j
(iii)t2i+1t2j= - i t2(i+j)+1
(iv)t2i+1t2j+1= 0
Proof: By our choice of V and W and homomorphism j, we see that the
induced map BV = BZ=p x BZ=p ! BW = BZ=p is the usual product on the
Hopf space BZ=p, which in turn gives us the usual product
: H*(B Z=p) H*(B Z=p) ! H*(B Z=p)
on the homology. The theorem follows (see chapter 2). 2
The theorem has the following interesting corollary.
Corollary 3.2.14Let m be an element of an unstable module M. When p = 2,
if tim = 0 and riis odd, then trm = 0. Now suppose p is odd. If t1m = 0,
then t2k+1m = 0. If t2im = 0 and riis not zero mod p, then t2rm = 0.
For emphasis' sake, realize that this implies that if t1m = 0, then t2k+1m = 0.
Chapter 4
uK"nneth relations
A classic problem in the study of homology and cohomology theories is finding
conditions under which one can compute E*(X x Y ) or E*(X x Y ) in terms of
E*X and E*Y or E*X and E*Y . In this chapter, we will examine situations
where
E*X E* E*Y ~=E*(X x Y )
which do not seem to be mentioned in other literature.
Let us restrict our attention to cohomology theories E* which have products
and satisfy the wedge axiom, i.e.
Y
E*(_ffXff) ~= E*(Xff):
ff
This allows us to reformulate many questions about cohomology theories into
questions about ring spectra ([Ad1 ] [Ad2 ] [Sw ]). A familiar theorem is
Theorem 4.0.1 [Ad1 ] [Ad2 ] [Sw ] If either E*X is a finitely generated free *
*right
E* module or E*Y is a finitely generated free left E* module, then
E*X E* E*Y ~=E*(X x Y ):
Another general situation which is known is the case E* = HF *, F a field,
and either HF *X or HF *Y is of finite type, i.e., each dimension contains only
a finite number of generators. If this is our case, then
HF *X F HF *Y ~=HF *(X x Y ):
Assuming, without loss of generality, that HF *Y is the one that is of finite
type, we claim that what is crucial about this last example is that
o HF *Y is a free module over HF *(pt) ~=F , and
CHAPTER 4. KU"NNETH RELATIONS 26
o HF *is bounded below, or coconnective: for all n less than a fixed integer,
0 in this case, and for all spaces W , HF nW = 0.
Specifically, we can show the following:
Theorem 4.0.2 Suppose E is a coconnective ring spectrum. If either E*X is
a free right E* module of finite type or E*Y is a free left E* module of finite
type, then
E*X E* E*Y ~=E*(X x Y ):
Hence by restricting the ring spectra we consider, we can loosen the require-
ment on the "finiteness" of E*X or E*Y .
To show the theorem, we need an algebraic lemma.
Lemma 4.0.3 If j 1 is finite and {Mff} is a collection of right modules over
a ring R, then Y Y
( Mff) R Rj ~= (MffR Rj)
Proof:
Y M Y
( Mff) R Rj ~= ( Mff) R R
j copies
~= M Y Mff
j copies
~= Y M Mff
j copies
~= Y MffR Rj
2
Proof of Theorem 4.0.2: The proof follows the style used to prove Theorem
4.0.1. Let us recall the style. Assume that E*Y is a free left E* module of fin*
*ite
type, the argument for E*X being a free right module of finite type is similar.
We regard Y as fixed and X as a variable. We have two functors
F1(-) = E*(-) E* E*Y; F2(-) = E*(- x Y )
Since E is a ring spectrum, we have the product map E^E ! E which induces a
natural transformation from F1 to F2. One then shows that each is a cohomology
theory satisfying the wedge axiom and that the two theories take on the same
values on a point, with the natural transformation inducing an isomorphism. If
E*Y is finitely generated, this is no problem. However, since tensor products
do not in general commute with products, we usually have trouble showing that
F1 satisfies the wedge axiom. However, E* is a coconnective theory, and this is
just enough to work.
CHAPTER 4. KU"NNETH RELATIONS 27
Q
We write E*Y as iNi, whereQNi is theQfree module on the generators of
E*Y in dimension i. Thus ( Mff) Ni~= (Mff Ni) by the previous lemma.
Let {Xff} be an arbitrary collection of spaces and let Mff= E*(Xff). So what
we need to show is that
Y Y
( Mff) E* E*Y ~= (MffE* E*Y ):
Q Q
There is a map from ( Mff) E* E*Y to (MffE* E*Y ), given by
((mff) n) = (mff n), where mff2 Mffand n 2 E*Y . We can define a
map going the other way as follows. For each i for which EiY is non-zero,
letQ{bi;r}r be a set of linearly independent generators for Ni. An element in
[ (MffE* E*Y )]k can be written in the form
X X
( mff;i;r bi;r);
i r
where mff;i;r2 Mk-iff. We would like to have
X X X X
(( mff;i;r bi;r)) = (mff;i;r bi;r):
i r i r
P P
It remains to show that the sum i r(mff;i;r bi;r) has only finite many non-
zero summands. Since for each i, {bi;r} is a finite set, it suffices to show th*
*at
there is only a finite number of i's such that mff;i;rcan be non-zero. Since we
are working with a coconnective theory, there is an integer N for which E*W
is zero in dimensions less than N. Thus there is a smallest value for i. The
coconnective hypothesis also implies that there is a smallest value for k - i
such that mff;i;rcan be non-zero, which in turnPimpliesPthat for each k, there
is a largest value of i. Therefore, the sum i r(mff;i;r bi;r) has only finite
many non-zero summands. One then observes that and are inverses of one
another and completes the proof as Theorem 4.0.1 is completed in [Sw ], [Ad1 ]
or [Ad2 ]. 2
Corollary 4.0.4If H*(Y ; R) is a free R module of finite type and R is finite,
then
H*(X x Y ; R) ~=H*(X; R) R H*(Y ; R):
Chapter 5
The integral cohomology
In this chapter, we reduce Conjecture 5.1.1 to a technical conjecture.
5.1 Bocksteins and H* (Map (B Z=p; X); Z)
We would like to show:
Conjecture 5.1.1 Suppose X is a space such that TV H*(X; Z=p) is of finite
type and the natural map (1.1) from TV H*(X; Z=p) to H*(Map (BV; X); Z=p),
induced by the evaluation map, is an isomorphism. If H*(X; Z=pi) is a free
Z=pi module, then H*(Map (BV; X); Z=pi) is a free Z=pi module.
Since TV = T dim V, it would suffice to show the theorem for V = Z=p. The
case i = 2 follows from Theorem 3.2.8, since fi1 is the 0th Milnor Bockstein.
Suppose we knew the result for all i less than some integer s. That would
imply that the first s - 1 Bocksteins are zero and that
H*(B Z=pxMap (B Z=p; X); Z=pi) ~=H*(B Z=p; Z=pi)H*(Map (B Z=p; X); Z=pi)
for all i less or equal to s (Corollary 4.0.4). We wish to show that fis is zer*
*o on
H*(Map (B Z=p; X); Z=ps) if fis is zero on H*(X; Z=ps).
We introduce some notation. Let "X : BZ=p x Map (B Z=p; X) ! X be the
evaluation map on X and let us abuse notation by letting "*X represent the
induced homomorphism from H*(X; R) to H*(B Z=p x Map (B Z=p; X); R) for
any commutative ring and also the composite homomorphism
H*(X; Z=pi) !
~=
H*(B Z=pxMap (B Z=p; X); Z=pi) !
H*(B Z=p; Z=pi) H*(Map (B Z=p; X); Z=pi)
CHAPTER 5. THE INTEGRAL COHOMOLOGY 29
for all i, 1 i s. The context will indicate which "*Xis meant.PFor a homoge-
nous element b 2 H*(X; Z=pi), denote its image under "*Xas xk tikb, where
xk is a generator of Hk(B Z=p; Z=pi) and tikb is an element of H*(Map (B Z=p; X*
*); Z=2i),
well-defined up to a multiple of an element of order p. We let and ae be defin*
*ed
by the exact sequence
0 ! Z=ps-1 ! Z=ps ae!Z=p ! 0
and define * and ae* to be the induced homomorphisms.
By Theorems 3.2.5 and 3.2.8, H*(Map (B Z=p; X); Z=p) is additively gen-
erated by elements of the form t2rm, where m belongs to a set of additive
generators for H*(X; Z=p). Thus, for i s, H*(Map (B Z=p; X); Z=pi) will
be generated by lifts of the trm's. Theorem 2.3.1 and our hypothesis tell us
that fis = 0 on the lifts of elements from H*(X; Z=p). Consider an arbitrary
element a 2 H*(X; Z=ps) which is a lift of some element ae*a in H*(X; Z=p).
If fisa = 0 , then its image under "*X, fis "*Xa, is zero, since all maps un-
der consideration are natural. One easily sees that "*Xa is a lift of "*Xae*a
(again,Pforgive the abuse of notation), and TheoremP3.2.8 implies that "*Xae*a =
vr t2rae*a (x2r if p = 2). Therefore, if "*Xa = kxk tska, ts2ra is a lift*
* of
t2rae*a and H*(Map (B Z=p; X); Z=pi) is generated by elements of the form ti2rb,
b 2 H*(X; Z=pi). P
Let us examine the consequences of fis (xk tska) = 0. Since fis is a
differential and additive, we have
X X
fis (xk tska)= fis(xk tska)
X
= (fisxk ae*tska + ae*xk fistska)
X
= (vr ae*ts2r-1a + vr fists2ra)
X
= vr (ae*ts2r-1a + fists2ra)
X
( = x2r (ae*ts2r-1fists2ra) if p =)2:
Examining the last sum term by term, we see that fists2ra = -ae*ts2r-1a for all*
* r.
Since a was a lift of an arbitrary element, the result would follow if ae*ts2r-*
*1a = 0
for all r and all a. We have assumed that H*(Map (B Z=p; X); Z=ps) is a free
Z=ps module, thus this last statement is equivalent to showing that tm2r-1a is a
multiple of p for all r and all a.
We now discuss some equivalents to this last condition. Choose a non-zero
element hr;s2 Hr(B Z=p; Z=ps) in dimension -r in the way that was discussed
in chapter 2. For any space X, define
tr;s: H*(X; Z=ps) ! H*-r(Map (B Z=p; X); Z=ps)
by tr;s(a) = "*X(a)=hr;s, where = denotes the slant product. (In the odd prime
case, this is uniquely defined only up to a unit multiple, i.e. choice of h1;s.*
* See
chapter 2.)
CHAPTER 5. THE INTEGRAL COHOMOLOGY 30
The operations tr;sare related to the tsroperations (when they exist) as
follows:
Lemma 5.1.2 tr;sa = ps-1tsra
Proof: Under our usual set of hypotheses, with H*(Map (B Z=p; X); Z=ps)
shown to be a free Z=ps module, we have
X
"*X(a)=hr;s= ( xk tska)=hr;s=< hr;s; xr > tsra:
Since < hr;s; xr >= ps-1 2 Z=ps, i.e. it has order p, tr;s(a) = ps-1 tsra.
2
Conjecture 5.1.3 Under the usual hypotheses, if fis 0 on H*(X; Z=ps), then
t2r-1;s 0.
The above discussion shows:
Theorem 5.1.4 Conjecture 5.1.3 implies Conjecture 5.1.1
Attempts to prove the Conjecture
To simplify the exposition, we limit ourselves to the prime 2; analogous state-
ments hold for odd primes.
The Bockstein homomorphism fis : HZ=2s ! HZ=2 is associated to the
short exact sequence
0 ! Z=2 ! Z=2s+1 ! Z=2s ! 0
and it has a lift to a Bockstein homomorphism "fis: HZ=2s ! HZ=2s associ-
ated to the short exact sequence
0 ! Z=2s ! Z=22s! Z=2s ! 0
with the lift given by the diagram
0 ! Z=2s ! Z=22s ! Z=2s ! 0
# # k
0 ! Z=2 ! Z=2s+1 ! Z=2s ! 0:
If fis 0 on H*(X; Z=2s) and a 2 H*(X; Z=2s) is an element of order 2s, then
we can show that
f"ist2r;s(a) = t2r-1;s(a) + t2r;s(f"is(a)) = t2r-1;s(a):
This would seem to suggest that we might be able to imitate the proof of
Theorem 3.2.8 to prove that t2r-1;s(a) = 0 for all r and all a 2 H*(X; Z=ps).
To do this, it would suffice to show the existence of a set of operations
Q k;s: HZ=2s ! |Qk;s|HZ=2s
with the properties that
CHAPTER 5. THE INTEGRAL COHOMOLOGY 31
(a) |Q k+1;s| > |Q k;s|
(b) Qk;s(ab) = (Q k;sa)b + a(Q k;sb)
(c) Qk;sh2r;s= h2r-|Qk;s|;s; Q k;sh2r-1;s= 0
which is equivalent to:
(c')Qk;sx2r-1= x2r-1+|Qk;s|; Q k;sx2r= 0
(d) If "fis 0 on the cohomology of a space X, then Qk;s 0 on the cohomol-
ogy of X.
We conclude this section with another conjecture, which is a corollary of
Conjecture 5.1.1. Let Zp represent the p-adic integers.
Conjecture 5.1.5 Suppose X is a space such that the natural map (1.1) from
TV H*(X; Z=p) to H*(Map (BV; X); Z=p), induced by the evaluation map, is an
isomorphism and suppose that TV H*(X; Z=p) is of finite type. If H*(X; Zp) is
a free Zp module, then H*(Map (BV; X); Zp) is a free Zp module.
5.2 Plausibility: Application to Lie Groups
In this section, we will provide further evidence Conjecture 5.1.1 is true. This
is done by examining two theorems, one by Dwyer and Zabrodsky, the other by
Borel.
To state Dwyer and Zabrodsky's result, we first need to make some defi-
nitions. A map f : X ! Y is defined to be a strong mod p equivalence if f
induces
~=
(a) an isomorphism ss0X ! ss0Y ,
~=
(b) an isomorphism ss1(X; x) ! ss1(Y; f(x)) for any x 2 X and
(c) an isomorphism
~=
H*(X"x; Z=p) ! H*(Y"f(x); Z=p);
where "Xxis the universal cover of the component of X containing x and
"Yf(x)is the universal cover of the component of Y containing f(x).
See [DZ ].
Consider the equivalence relation ~ on the set of group homomorphisms
from ss to G given by j ~ if there is an h 2 G such that j(a) = h(a)h-1 for
all a 2 ss. Let Rep(ss; G) be the set of equivalence classes under this relatio*
*n. If
we let ZG (j) denote the centralizer in G of the image of j, then we have:
CHAPTER 5. THE INTEGRAL COHOMOLOGY 32
Theorem 5.2.1 ([DZ ]) If ss is a finite p-group and G is a compact Lie group,
then there is a natural map
a
f : BZG (j) ! Map (Bss; BG)
j2Rep(ss;G)
which is a strong mod p equivalence.
An elementary abelian p subgroup of a compact Lie group G is called toral
if it can be embedded in a torus of G. Borel's theorem follows.
Theorem 5.2.2 ([Bo ])Let G be a connected compact Lie group and let BG
be its classifying space. The following are equivalent:
a) The integral cohomology of G has no p torsion.
b) The integral cohomology of BG has no p torsion.
c) Every elementary abelian p group in G is toral.
d) Every elementary abelian p group of rank 3 in G is toral.
If G is such a Lie group, certain subgoups of G will inherit property c; amo*
*ng
these are the centralizers of elements of order p. Hence we have the following
theorem.
Theorem 5.2.3 Let G be a compact Lie group whose integral cohomology is p-
torsion free and of finite type. If V is an elementary abelian p-group and j is*
* a
homorphism from V to G, then H*(BZG (j); Z) is p-torsion free. In particular,
if g is an element of order p, then H*(BZG (g); Z) is p-torsion free.
This theorem, combined with Theorem 5.2.1, implies that Conjecture 5.1.5
holds for connected compact Lie groups.
We remark that Borel's proof of his theorem uses the classification of com-
pact Lie groups.
Chapter 6
Speculations
6.1 Connective Morava K-theory
The style of proof proposed for Conjecture 5.1.1 suggests that there may sim-
ilar results with other cohomology theories. An example is the nth connective
Morava k-theory k(n), n > 0 [JW ] [WSW ]. The coefficient ring of k(n) is Fp[v*
*n],
where the dimension of vn is -2(pn - 1). There is a fibration
n-1) vn
2(p k(n) ! k(n) ! HZ=p
of spectra which gives rise to a Bockstein spectral sequence much like the or-
dinary mod p Bockstein spectral sequence. In studying this spectral sequence,
one sees fibrations n r
2r(p -1)k(n) vn!k(n) ! Er;
where the theory associated to Er has the coefficient ring E*r~=Fp[vn]=(vrn).
As before, we assume that X is a space such that the natural map from
T H*(X; Z=p) to H*(Map (BZ=p; X); Z=p) is an isomorphism.
Conjecture 6.1.1 If E*r(X) is a free module over the coefficient ring E*r, then
E*r(Map (BZ=p; X)) is also a free module over E*r.
The conjecture is true for r = 1, as E1 = HZ=p. I can show it is true for
r = 2 (this is a corollary of Theorem 3.2.8 in my summary). The proof should
be analogous to the proof of Theorem 5.1.1. If the conjecture is true, some
version of the following conjecture would seem to be a corollary.
Conjecture 6.1.2 If k(n)*(X) is a free module over the ring k(n)*, then so is
k(n)*(Map (BZ=p; X)).
These conjectures motivate the style of arguments and proofs in much of this
dissertation. The generality of the statement of Theorem 3.2.8 and the spectra
style proof of Theorem 4.0.2 are examples of this.
CHAPTER 6. SPECULATIONS 34
6.2 The cohomology of Map (B Z=p; X)f
A map f : B Z=p ! X lies in some path component of Map (B Z=p; X); we
call this component Map (B Z=p; X)f. The T functor technology provides us
a way to compute the mod p cohomology of Map (B Z=p; X)f under our usual
hypothesis that the natural map j defined in chapter 3 from T H*(X; Z=p) to
H*(Map (B Z=p; X); Z=p) is an isomorphism.
The map f or any other map in Map (B Z=p; X)f induces a homomorphism
f* : H*(X; Z=p) ! H*(B Z=p; Z=p), unique since Map (B Z=p; X)f is path con-
nected and a path in this space of maps is a homotopy between the two maps
which are the endpoints. Since H*(B Z=p; Z=p) ~= H*(B Z=p; Z=p) Fp, f*
induces a map T H*(X; Z=p) ! Fp. The last map is actually a map from
(T H*(X; Z=p))0 to Fp, which we will label f". Defining TfH*(X; Z=p) to be
T H*(X; Z=p) R Fp, where R = (T H*(X; Z=p))0 and the action across the
tensor product is provided by "f, it follows that the natural map j induces a
map
jf : TfH*(X; Z=p) ! H*(Map (B Z=p; X)f; Z=p):
Furthermore, if j is an isomorphism, so is jf.
A natural question that arises is that if Conjecture 5.1.1 and the other con-
jectures are true, are there similar statements with more restrictive hypotheses
which yield similar results for the path component Map (B Z=p; X)f?
Two preliminary steps in this direction are provided by the next two results.
The elements in T H*(X)0
Proposition 6.2.1Let X be a space and f : BZ=p ! X be a map. We have
that "f(t|m|m) is non-zero if and only if f*(m) 2 H*(B Z=p) is non-zero.
Proof: The map
f* : H*(X) ! H*(B Z=p) ~=H*(B Z=p) Fp
sends m to the element f*(m) 1. Recalling how Adams showed the existence
of T (see chapter 3), we see that f* induces a map
H*(B Z=p) H*(X) ! Fp
which sends h|m| m to 1. The proposition follows. 2
Therefore we have a beachhead for our questions.
Dwyer-Wilkerson's approach
W. Dwyer and C. Wilkerson in [DW ] have shown another way of computing
TfH*(X) that the author is not yet sure how to make compatible with his
approach of viewing T locally. However, using Dwyer-Wilkerson's approach,
CHAPTER 6. SPECULATIONS 35
we can make a statement like Theorem 3.2.8. Let us examine their approach,
which seems to be inspired by Smith theory. We will restrict our discussion to
the immediately applicable cases. See [DW ] for the more general statements.
Let f be a map that as before maps from B Z=p to X. The ring homo-
morphism f* : H*(X; Z=p) ! H*(B Z=p; Z=p) provides us a way of making
H*(B Z=p; Z=p) into a (left) module over H*(X; Z=p), with the structure given
by a . b = f*(a)b. Now define Sf to be the multiplicative set generated by those
elements in H2(X; Z=p) that are in the image of the Bockstein fi, (Sq1 ifp = 2)
of those elements of H1(X; Z=p) which map nonzero under f*. Let I = kerf*.
Theorem 6.2.2 [DW ] If H*(B Z=p; Z=p) is finitely generated as a module over
H*(X; Z=p), then
TfH*(X; Z=p) ~=Un ((S-1fH*(X; Z=p))^I)
where Un M is the largest unstable submodule of M for M a module over Ap.
Suppose we wish to know when the Bockstein fi will be zero on such a
TfH*(X; Z=p). We see from the equation
fim = fi(sm=s) = (fis)m=s + s(fim=s) = s(fim=s);
that fim=s = (fim)=s which will equal zero (0=1) when there is a sm 2 Sf such
that sm fim = 0. If for all m 2 H*(X; Z=p) there exists such a sm 2 Sf, then
fi = 0 on TfH*(X; Z=p).
For more on: localizations and completions of rings, see [Mat ]; the action *
*of
the Steenrod algebra on the localizations and completions, see [Si] and [Wi ].
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