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universal coefficient theorem

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Idea

The universal coefficient theorem states how ordinary homology/ordinary cohomology determined homology/cohomology with arbitrary coefficients.

For C C_\bullet a chain complex (of abelian groups) and FF a field (the coefficient field), the homology group H p(C,F)H_p(C,F) and the cohomology group H p(C,F)H^p(C,F) are indeed related by dualization: H p(C,F)Hom F(H p(C,F),F) H^p(C,F) \simeq Hom_F(H_p(C,F), F). If the coefficients are not a field but an arbitrary abelian group, then this relationship receives a correction by an Ext-group. This is discussed below in For ordinary cohomology.

Dually, again if FF is a field then there is an isomorphism H n(C)FH n(CF)H_n(C) \otimes F \simeq H_n(C \otimes F) and for more general FF this is corrected by a Tor-group. This is discussed below in For ordinary homology.

More generally, under suitable conditions there are universal coefficient theorems that relate generalized (Eilenberg-Steenrod) cohomology to the dual of generalized homology. This is discussed below in For generalized cohomology.

There is also a version of the theorem for Kasparov’s KK-theory, see the references.

Statement

For ordinary cohomology

Let C C_\bullet be a chain complex of free abelian groups. Let AA be an arbitrary abelian group.

Write

  • C Hom Ab(C ,A)C^\bullet \coloneqq Hom_{Ab}(C_\bullet, A) for the dual cochain complex with respect to AA;

  • H n(C)H_n(C) for the chain homology of C C_\bullet

  • H n(C,A)H^n(C,A) for the cochain cohomology of C C^\bullet hence for the cochain cohomology of C C_\bullet with coefficients in AA.

Remark

There is a canonical morphism of abelian groups

()():H n(C,A)Hom Ab(H n(C,),A) \int_{(-)}(-) : H^n(C,A) \to Hom_{Ab}(H_n(C,\mathbb{Z}), A)

given by sending a cocycle to evaluation of that cocycle on a chain:

[ω]([σ] σωω(σ)) [\omega] \mapsto \left( [\sigma] \mapsto \int_{\sigma} \omega \coloneqq \omega(\sigma) \right)
Theorem

(universal coefficient theorem in ordinary cohomology)

The morphism hh is surjective and its kernel is the Ext group Ext 1(H n1(C,),A)Ext^1(H_{n-1}(C, \mathbb{Z}), A). In other words, there is a short exact sequence

0Ext 1(H n1(C),A)H n(C,A)Hom Ab(H n(C),A)0 0 \to Ext^1(H_{n-1}(C), A) \to H^n(C, A) \to Hom_{Ab}(H_n(C), A) \to 0

hence

0Ext 1(H n1(C),A)H n(Hom Ab(C ,A))Hom Ab(H n(C),A)0 0 \to Ext^1(H_{n-1}(C), A) \to H^n(Hom_{Ab}(C_\bullet,A)) \to Hom_{Ab}(H_n(C), A) \to 0

Moreover, this sequence splits (non-canonically).

We reproduce the direct proof given for instance in (Boardman).

Lemma

Given a homomorphism A 1fA 2gA 3A_1 \stackrel{f}{\to} A_2 \stackrel{g}{\to} A_3 of abelian groups together with a retract s:A 3A 2s : A_3 \to A_2 of gg, there is a short exact sequence of cokernels

0cokerfgcoker(gf)coker(g)0. 0 \to coker f \stackrel{g'}{\to} coker(g \circ f) \to coker(g) \to 0 \,.
Proof

Since we work in Ab, all the cokernels appearing here (as discussed there) may be expressed as quotients, e.g coker(f)A 2/im(f)coker(f) \simeq A_2/im(f).

The sequence of inclusions im(gf)im(g)A 3im(g \circ f) \hookrightarrow im(g) \hookrightarrow A_3 induces the canonical short exact sequence

0im(g)im(gf)A 3im(gf)A 3im(g)0 0 \to \frac{im(g)}{im(g \circ f)} \to \frac{A_3}{im(g \circ f)} \to \frac{A_3}{im(g)} \to 0

and we claim that this is already isomorphic to the one stated in the lemma. This is manifestly true for the two terms on the right. For the term on the left observe that gg induces a morphism g:A 2/im(f)A 3/im(gf)g' : A_2 / im(f) \to A_3 / im(g \circ f). By the existence of the retract ss this has itself a retract. Moreover it factors as

g:A 1/im(f)im(g)/im(gf)A 3/im(gf). g' : A_1/im(f) \to im(g)/im(g \circ f) \hookrightarrow A_3/ im(g \circ f) \,.

Therefore the first morphism here on the left has to be an isomorphism, too.

Proof (of theorem 1)

Write

0B nZ nH n0 0 \to B_n \to Z_n \to H_n \to 0

for the short exact sequence of boundaries, cycles, and homology groups of C C_\bullet in degree nn. Since C nC_n is assumed to be a free abelian group and since B nB_n and Z nZ_n are subgroups, it follows that these are also free abelian, by the abelian Nielsen-Schreier theorem. Therefore this sequence exhibits a projective resolution of the group H nH_n. It follows that the Ext-group Ext 1(H n,A)Ext^1(H_n,A) is characterized by the short exact sequence

(1)Hom(Z n,A)Hom(B n,A)Ext 1(H n,A)0. Hom(Z_n, A) \to Hom(B_n,A) \to Ext^1(H_n,A) \to 0 \,.

Notice also that the short exact sequence

(2)0Z nC nB n10 0 \to Z_n \to C_n \stackrel{\partial}{\to} B_{n-1} \to 0

is split because, as before, B n1B_{n-1} is free abelian. Using these two exact sequences on the left and right of the short exact sequence

0Z n/B nC n/B nC n/Z n0 0 \to Z_n/B_n \to C_n/B_n \to C_n/Z_n \to 0

shows that this is equivalent to

(3)0H nC n/B nB n1. 0 \to H_n \to C_n/B_n \stackrel{\partial}{\to} B_{n-1} \,.

Again this splits as B n1B_{n-1} is free abelian.

In addition to these exact sequence consider the decomposition

:C nC n/B nC n/Z nB n1Z n1C n1 \partial : C_n \to C_n/B_n \to C_n/Z_n \stackrel{\simeq}{\to} B_{n-1} \hookrightarrow Z_{n-1} \hookrightarrow C_{n-1}

and apply Hom(,A)Hom(-,A) to obtain the diagram

0 Hom(H n,A) Hom(B n,A) Hom(C n,A) i Hom(C n/B n,A) 0 0 Hom(¯,A) 0 Ext 1(H n,A) Hom(B n1,A) Hom(Z n1,A) 0 Hom(C n1,A) \array{ && && 0 \\ && && \uparrow \\ && && Hom(H_n,A) \\ && && \uparrow \\ Hom(B_n,A) &\leftarrow& Hom(C_n,A) &\stackrel{i}{\leftarrow}& Hom(C_n/B_n,A) &\leftarrow& 0 && 0 \\ && && \uparrow^{\mathrlap{Hom(\bar \partial,A)}} && && \uparrow \\ 0 &\leftarrow& Ext^1(H_n,A) &\leftarrow& Hom(B_{n-1},A) && \leftarrow && Hom(Z_{n-1},A) \\ && && \uparrow && \nwarrow && \uparrow \\ && && 0 && && Hom(C_{n-1},A) }

Here the right vertical sequence is exact, because (2) splits, and the left vertical sequence is exact because (3) splits. The upper horizontal sequence is exact because the hom functor takes cokernels to kernels and finally the lower horizontal sequence is the exact sequence (1).

Since therefore ii and Hom(¯,A)Hom(\bar \partial,A) are monomorphisms, it follows that the degree nn-cocycles are

Z n1ker(Hom(C n1,A)Hom(C n,a))ker(Hom(C n1,A)Hom(B n1,A)). Z^{n-1} \coloneqq ker( Hom(C_{n-1},A) \to Hom(C_n,a) ) \simeq ker( Hom(C_{n-1},A) \to Hom(B_{n-1}, A) ) \,.

Using this for n1n-1 replaced by nn shows by the upper horizontal exact sequence that

Z n=Hom(C n/B n,A). Z^n = Hom( C_n/B_n, A) \,.

Similarly the coboundaries are seen to be

B nimHom(,A)im(Hom(Z n1,A)Hom(C n/B n),A). B^n \coloneqq im Hom(\partial,A) \simeq im ( Hom(Z_{n-1}, A) \to Hom(C_n/B_n), A ) \,.

Together this gives the cochain cohomology as

H n(C,A)Z n/B ncoker(Hom(Z n,A)Hom(C n/B n,A)). H^n(C,A) \coloneqq Z^n / B^n \simeq coker ( Hom(Z_n, A) \to Hom( C_n/B_n, A ) ) \,.

Now the universal coefficient theorem follows by going into lemma 1 with the identifications A 1=Hom(Z n1,A)A_1 = Hom(Z_{n-1}, A), A 2=Hom(B n1,A)A_2 = Hom(B_{n-1}, A), A 3=Hom(C n/B n,A)A_3 = Hom(C_n/B_n,A).

For ordinary homology

Let C Ch (Ab)C_\bullet \in Ch_\bullet(Ab) be a chain complex of free abelian groups, and let AA \in Ab be any abelian group. Write C AC_\bullet \otimes A etc. for the degreewise tensor product of abelian groups.

More generally, let RR be a ring which is a principal ideal domain (in the above R=R = \mathbb{Z} is the ring of integers), let C Ch (RMod)C_\bullet \in Ch_\bullet(R Mod) be a chain complex of free modules over RR, let ARModA \in R Mod be any RR-module and write C k RAC_k \otimes_R A for the tensor product of modules over RR.

Theorem

For each nn \in \mathbb{N} there is a short exact sequence

0H n(C ) RAH n(C RA)Tor 1 RMod(H n1(C ),A)0 0 \to H_n(C_\bullet) \otimes_R A \to H_n(C_\bullet \otimes_R A) \to Tor_1^{R Mod}(H_{n-1}(C_\bullet), A) \to 0 \,

where on the right we have the first Tor-module of the chain homology H n1(C )H_{n-1}(C_\bullet) with AA.

Remark

A fairly direct generalization of this statement and its proof is the Künneth theorem in ordinary homology, see at Künneth theorem - In ordinary homology.

We spell out a proof along the lines for instance given in (Hatcher, 3.A) or (Chen, section 3).

Lemma

For C C_\bullet a chain complex of free abelian groups and AA \in Ab any abelian group, there is a long exact sequence of the form

B nAi nAZ nAH n(C A)B n1Ai n1AZ n1A, \cdots \to B_n \otimes A \stackrel{i_n \otimes A}{\to} Z_n \otimes A \to H_n(C_\bullet \otimes A) \to B_{n-1} \otimes A \stackrel{i_{n-1}\otimes A}{\to} Z_{n-1} \otimes A \to \cdots \,,

where B nB_n are the boundaries and Z nZ_n the cycles of C C_\bullet in degree nn and where i n:B nZ ni_n \colon B_n \hookrightarrow Z_n is the canonical inclusion.

Proof

Since, by the Dedekind-Nielsen-Schreier theorem, every subgroup of a free abelian group is itself free abelian, such as the subgroup of cycles Z nC nZ_n \hookrightarrow C_n, it follows that for each nn \in \mathbb{N} we have a splitting of the short exact sequence 0Z nC nB n10 \to Z_n \to C_n \to B_{n-1} and hence (as discussed at split exact sequence) a direct sum decomposition

C nZ nB n1. C_n \simeq Z_n \oplus B_{n-1} \,.

Here the second direct summand on the right identifies under the differential C\partial^C with the boundaries in one degree lower, since by construction C\partial^C is injective on C n/Z nC_n/Z_n.

Accordingly, if we regard the graded abelian groups B B_\bullet and Z Z_\bullet as chain complexes with vanishing differential, then we have a sequence of chain maps

0Z C B 10 0 \to Z_\bullet \hookrightarrow C_\bullet \to B_{\bullet-1} \to 0

which is degreewise a short exact sequence, hence is a short exact sequence of chain complexes. Now since the tensor product of abelian groups distributes over direct sum, the image of this sequence under ()A(-)\otimes A

0Z AC AB 1A0 0 \to Z_\bullet \otimes A \hookrightarrow C_\bullet \otimes A \to B_{\bullet-1} \otimes A \to 0

is still a short exact sequence. The induced homology long exact sequence, as discussed there, is the long exact sequence to be shown.

Proof

of theorem 2

By lemma 2 we have short exact sequences

0coker(i nA)H n(C A)ker(i nA)0 0 \to coker(i_n \otimes A) \to H_n(C_\bullet \otimes A) \to ker(i_n \otimes A) \to 0

Since the tensor product of abelian groups is a right exact functor it preserves cokernels and hence

coker(i nA)coker(i n)A=H n(C)A. coker(i_n \otimes A) \simeq coker(i_n) \otimes A = H_n(C) \otimes A \,.

The dual statement were true if ()A(-)\otimes A were also a left exact functor. In general it is not, and the failure is measure by the Tor-group:

Notice that by assumption and by the Dedekind-Nielsen-Schreier theorem the defining short exact sequence

0B ni nZ nH n(C )0 0 \to B_n \stackrel{i_n}{\to} Z_n \to H_n(C_\bullet) \to 0

exhibits [0B nZ n] qiH n(C)[\cdots \to 0 \to B_n \to Z_n] \stackrel{\simeq_{qi}}{\to} H_n(C) as a projective resolution of H n(C )H_n(C_\bullet). Therefore by definition of Tor the group Tor 1(H n(C ),A)Tor_1(H_n(C_\bullet), A) is the chain homology in degree 1 of

[0B nGi nAZ nA], [\cdots \to 0 \to B_n \otimes G \stackrel{i_n \otimes A}{\to} Z_n \otimes A] \,,

which is

Tor 1(H n(C ),A)ker(i nA). Tor_1(H_n(C_\bullet), A) \simeq ker(i_n \otimes A) \,.

For generalised cohomology theories

The situation for generalised cohomology theories is much more complicated than that for ordinary cohomology due to the fact that it is harder (or impossible!) to use the tools of chain complexes. Nonetheless, it is possible to say something. The general case was studied by Adams in Ada69 and the initial version of the rest of this section is heavily based on that treatment. This was also considered in the slightly later work, Ada74, Chapter 13. Adams’ opening paragraph in Ada69 is worth quoting in its entirety as motivation for this study.

It is an established practice to take old theorems about ordinary homology, and generalise them so as to obtain theorems about generalised homology theories. For example, this works very well for duality theorems about manifolds. We may ask the following question. Take all those theorems about ordinary homology which are standard results in every day use. Which are the ones which still lack a fully satisfactory generalisation to generalised homology theories? I want to devote this lecture to such problems.

J. F. Adams

The lecture concentrates on the Universal Coefficient Theorem and, as a by-product, the Künneth theorem.

Let E *E^* and F *F^* be two generalized cohomology theories and E *E_* and F *F_* two generalized homology theories, such that EE is multiplicative and FF is a module over EE. Then the general problems that a Universal Coefficient Theorem should apply to are the following:

  1. Given E *(X)E_{*}(X), calculate F *(X)F_{*}(X).

  2. Given E *(X)E_{*}(X), calculate F *(X)F^{*}(X).

  3. Given E *(X)E^{*}(X), calculate F *(X)F_{*}(X).

  4. Given E *(X)E^{*}(X), calculate F *(X)F^{*}(X).

In Ada69, Adams works in a very general setting. On this page, we shall work in a more restricted situation (as spelled out in Note 2 in Adams’ lectures). We assume that E *()E^{*}(-) is the generalised cohomology theory associated to a commutative ring spectrum, EE. The cohomology theory F *()F^{*}(-) is assumed to come from a left module-spectrum over EE, which we shall denote by FF. We do not assume that FF is itself a ring spectrum. Following Adams, we shall also assume that all our cohomology and homology theories are reduced.

There are two statements that one would like to hold. These are not themselves theorems, rather the theorem would say ‘‘Under certain conditions, these statements hold’’. The statements are the following.

UCT1

There is a spectral sequence

Tor p,* E *(E *(X),F *)pF *(X) \Tor_{p,*}^{E_{*}} (E_{*}(X), F_{*}) \xRightarrow[p]{} F_{*}(X)

with edge homomorphism

E *(X) E *F *F *(X). E_{*}(X) \otimes _{E_{*}} F_{*} \to F_{*}(X).
UCT2

There is a spectral sequence

Ext E * p,*(E *(X),F *)pF *(X) \Ext_{E_{*}}^{p,*} (E_{*}(X), F^{*}) \xRightarrow[p]{} F^{*}(X)

with edge homomorphism

F *(X)Hom E *(E *(X),F *). F^{*}(X) \to \Hom_{E_{*}} (E_{*}(X), F^{*}).

For finite CW-complexes then we can derive two further statements from the above by S-duality. We use the notation DXD X for the Spanier-Whitehead dual of XX.

For a finite CW-complex XX, we can apply UCT1 and UCT2 to DXD X in place of XX and then use the various isomorphisms relating the cohomologies of XX and DXD X to reformulate them in terms of XX. We thus get the following statements.

UCT3

For XX a finite CW-complex, there is a spectral sequence

Tor p,* E *(E *(X),F *)pF *(X) \Tor_{p,*}^{E^{*}}(E^{*}(X), F^{*}) \xRightarrow[p]{} F^{*}(X)

with edge homomorphism

E *(X) E *F *F *(X). E^{*}(X) \otimes _{E^{*}} F^{*} \to F^{*}(X).
UCT4

For XX a finite CW-complex, there is a spectral sequence

Ext E * p,*(E *(X),F *)pF *(X) \Ext^{p,*}_{E^{*}} (E^{*}(X), F_{*}) \xRightarrow[p]{} F_{*}(X)

with edge homomorphism

F *(X)Hom E * *(E *(X),F *). F_{*}(X) \to \Hom^{*}_{E^{*}}(E^{*}(X), F_{*}).

A particularly important special case of these statements is when we have a topological space, say YY, and a cohomology theory, E *()E^{*}(-). Then we define a new homology theory F *()F_{*}(-) by F *(X)=E *(XY)F_{*}(X) = E_{*}(X \wedge Y) and a new cohomology theory G *()G^{*}(-) by G *(X)=E *(XY)G^{*}(X) = E^{*}(X \wedge Y). These are representable, the homology theory by YEY \wedge E and the cohomology theory by the function spectrum F(Y,E)F(Y,E). Putting these into the statements of the universal coefficient theorem, we obtain similar statements for the Künneth theorem.

KT1

There is a spectral sequence

Tor p,* E *(E *(X),E *(Y))pE *(XY) \Tor_{p,*}^{E_{*}} (E_{*}(X), E_{*}(Y)) \xRightarrow[p]{} E_{*}(X \wedge Y)

with edge homomorphism

E *(X) E *E *(Y)E *(XY). E_{*}(X) \otimes _{E_{*}} E_{*}(Y) \to E_{*}(X \wedge Y).
KT2

There is a spectral sequence

Ext E * p,*(E *(X),E *(Y))pE *(XY) \Ext_{E_{*}}^{p,*}(E_{*}(X), E^{*}(Y)) \xRightarrow[p]{} E^{*}(X \wedge Y)

with edge homomorphism

E *(XY)Hom E * *(E *(X),E *(Y)). E^{*}(X \wedge Y) \to \Hom_{E_{*}}^{*} (E_{*}(X), E^{*}(Y)).
KT3

For XX a finite CW-complex, there is a spectral sequence

Tor p,* E *(E *(X),E *(Y))pE *(XY) \Tor_{p,*}^{E^{*}} (E^{*}(X), E^{*}(Y)) \xRightarrow[p]{} E^{*}(X \wedge Y)

with edge homomorphism

E *(X) E *E *(Y)E *(XY). E^{*}(X) \otimes _{E^{*}} E^{*}(Y) \to E^{*}(X \wedge Y).
KT4

For XX a finite CW-complex, there is a spectral sequence

Ext E * p,*(E *(X),E *(Y))pE *(XY) \Ext^{p,*}_{E^{*}} (E^{*}(X), E_{*}(Y)) \xRightarrow[p]{} E_{*}(X \wedge Y)

with edge homomorphism

E *(XY)Hom E * *(E *(X),E *(Y)). E_{*}(X \wedge Y) \to \Hom_{E^{*}}^{*} (E^{*}(X), E_{*}(Y)).

The key question is, thus: when do these statements hold? Adams gives some answers in Ada69.

A Special Case

In both Ada69 and Ada74, there is a particular focus on the universal coefficient theorem coming from its applications to the Adams spectral sequence. With that aim in mind, he studies the universal coefficient theorems with considerably strong assumptions. These assumptions are designed to allow Atiyah’s method (from Ati62) to work.

Assumption

(Condition 13.3 in Ada74, see also Assumption 20 in Ada69).

The spectrum EE is the direct limit of finite spectra E αE_\alpha for which E *(DE α)E_*(D E_\alpha) is projective over E *E_* and

F *(DE α)Hom E * *(E *(DE α),F *) F^*(D E_\alpha) \to \Hom^*_{E_*}(E_*(D E_\alpha), F_*)

is an isomorphism for all module-spectra FF over EE. Here, DE αD E_\alpha is the S-dual of E αE_\alpha.

The main difference between the two treatments is that in Ada69, the condition involving FF is stated for a single module-spectrum, not for all module-spectra, and there are alternatives for homology (F *()F_*(-)) and cohomology (F *()F^*(-)).

In the comments following Assumption 20 in Ada69, Adams remarks that this is implied by a stronger condition (Proposition 17) on EE which makes no reference to FF. As EE is a ring spectrum, this reduces to:

  1. The spectral sequence H *(E α;E *)E *(E α)H^*(E_\alpha; E^*) \Rightarrow E^*(E_\alpha) is trivial, and
  2. For each pp, H p(E α;E *)H^p(E_\alpha; E^*) is projective as an E *E^*-module.

With this assumption, Adams shows the following result:

Theorem

Let EE be a ring spectrum satisfying the assumption above. Let FF be a module-spectrum over EE. Then 2 holds, and the spectral sequence is convergent.

What “convergent” means here is spelled out in Ada74, Theorem 8.2.

Corollary

Let EE be a ring spectrum satisfying the assumption above. Suppose that E *(X)E_*(X) is projective over E *E_*. Then the spectral sequence from 2 collapses at the E 2E^2 term. That is,

F *(X)Hom E * *(E *(X),F *) F^*(X) \to \Hom^*_{E_*}(E_*(X),F_*)

is an isomorphism.

In Ada74, Adams lists several cohomology theories (for E *()E^*(-)) where the assumption holds. These are: SS, H pH\mathbb{Z}_p, MOMO, MUMU, MSpMSp, KK, KOKO.

Freeness and Flatness

In BJW95 and Boa95 there are various versions of the universal coefficient theorems and Künneth theorems which are stated and proved (or indications given on how to prove) under assumptions of either freeness or flatness.

Here, we shall gather together all the statements made. In all the following, E *()E^*(-) is a multiplicative generalised cohomology theory with representing ring spectrum EE. We use E *()E_*(-) for the associated homology theory. Following Boa95 and BJW95, cohomology and homology are not reduced in this section.

Theorem (Boa95, 4.2)

Assume that E *(X)E_*(X) or E *(Y)E_*(Y) is a free or flat E *E^*-module. Then the pairing:

×:E *(X)E *(Y)E *(X×Y), \times \colon E_*(X) \otimes E_*(Y) \to E_*(X \times Y),

induces the Künneth isomorphism E *(X×Y)E *(X)E *(Y)E_*(X \times Y) \cong E_*(X) \otimes E_*(Y) in homology.

The next result relates homology and cohomology.

Theorem (Boa95, 4.14)

Assume that E *(X)E_*(X) is a free E *E^*-module. Then XX has strong duality, i.e. the duality map d:E *(X)DE *(X)d \colon E^*(X) \to D E_*(X) is a homeomorphism between the profinite topology on E *(X)E^*(X) and the dual-finite topology on DE *(X)D E_*(X). In particular, E *(X)E^*(X) is complete Hausdorff.

Combining these two gives the Künneth theorem for cohomology.

Theorem (Boa95, 4.19)

Assume that E *(X)E_*(X) and E *(Y)E_*(Y) are free E *E^*-modules. Then we have the Künneth homeomorphism E *(X×Y)E *(X)^E *(Y)E^*(X \times Y) \cong E^*(X) \widehat{\otimes} E^*(Y) in cohomology.

There are similar results for spectra. Boardman, Johnson, and Wilson write reduced homology and cohomology as E *(X,o)E_*(X,o) and E *(X,o)E^*(X,o), even when XX is a spectrum (and so the reduced theories are all that there are).

Theorem (Boa95, 9.20)

Assume that E *(X,o)E_*(X,o) or E *(Y,o)E_*(Y,o) is a free or flat E *E^*-module. Then the pairing ×:E *(X,o)E *(Y,o)E *(XY,o)\times \colon E_*(X,o) \otimes E_*(Y,o) \to E_*(X \wedge Y,o) is an isomorphism in homology.

Theorem (Boa95, 9.25)

Assume that E *(X,o)E_*(X,o) is a free E *E^*-module. Then XX has strong duality, i.e. d:E *(X,o)DE *(X,o)d \colon E^*(X,o) \to D E_*(X,o) is a homeomorphism between the profinite topology on E *(X,o)E^*(X,o) and the dual-finite topology on DE *(X,o)D E_*(X,o). In particular, E *(X,o)E^*(X,o) is complete Hausdorff.

Theorem (Boa95, 9.31)

Assume that E *(X,o)E_*(X,o) and E *(Y,o)E_*(Y,o) are free E *E^*-modules. Then the pairing

×:E *(X,o)^E *(Y,o)E *(XY,o) \times \colon E^*(X,o) \widehat{\otimes} E^*(Y,o) \to E^*(X \wedge Y,o)

induces the cohomology Künneth homeomorphism.

Examples

For singular cohomology

For XX a topological space, write SingXSing X for its singular simplicial complex and

C (X)N[SingX] C_\bullet(X) \coloneqq N \mathbb{Z}[Sing X]

for the normalized chain complex of the simplicial abelian group obtained by forming degreewise the free abelian group.

The singular homology H (X)H_\bullet(X) of XX is the chain homology of C (X)C_\bullet(X), and for AA some coefficient abelian group, the singular cohomology H (X,A)H^\bullet(X,A) is the cochain cohomology, of C (X)C_\bullet(X) with coefficients in AA.

Comparison with the ordinary universal coefficient theorem 1 shows that:

Theorem

(universal coefficient theorem in topology)

For XX a topological space, AA an abelian group and n1n \geq 1 \in \mathbb{N}, the singular homology and singular cohomology of XX fit into a split short exact sequence of the form

0Ext 1(H n1(X))H n(X,A)Hom Ab(H n(X),A)0. 0 \to Ext^1(H_{n-1}(X)) \to H^n(X,A) \to Hom_{Ab}(H_n(X), A) \to 0 \,.

References

For ordinary (co)homology

An exposition of the universal coefficient theorem for ordinary cohomology and homology is in section 3.1 of

The note

  • Adam Clay, The universal coefficient theorems and Künneth formulas (pdf)

surveys and spells out statement and proof of the theorem. A detailed proof of the theorem in cohomology is also in

and a detailed proof of the statement in homology is in section 3 of

  • Chen, Universal coefficient theorem for homology (pdf)

For generalized (co)homology

Universal coefficient theorems for generalized homology are discussed in

  • Friedrich Bauer, Remarks on universal coefficient theorems for generalized homology theories Quaestiones Mathematicae Volume 9, Issue 1 & 4, 1986, Pages 29 - 54

More on the universal coefficient theorem in generalized homology is in:

  • Ada69 J. F. Adams. Lectures on generalised cohomology. pages 1–138, Berlin, 1969. Springer.

  • Ada74 J. F. Adams, (1974). Stable homotopy and generalised homology. Chicago, Ill.: University of Chicago Press.

  • Ati62 M. F. Atiyah. Vector bundles and the Künneth formula. Topology, 1:245–248, 1962.

  • Boa95 J. M. Boardman, (1995). Stable operations in generalized cohomology. (pp. 585–686). Amsterdam: North-Holland.

  • BJW95 J. M. Boardman, and Johnson, David Copeland and Wilson, W. Stephen. (1995). Unstable operations in generalized cohomology. (pp. 687–828). Amsterdam: North-Holland.

For KK-theory

Discussion for KK-theory is in

  • Jonathan Rosenberg, Claude Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. Volume 55, Number 2 (1987), 431-474. (EUCLID)

Revised on August 7, 2013 17:51:06 by Urs Schreiber (131.174.43.103)