group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
and
nonabelian homological algebra
The universal coefficient theorem states how ordinary homology/ordinary cohomology determines homology/cohomology with arbitrary coefficients.
For $C_\bullet$ a chain complex (of abelian groups) and $F$ a field (the coefficient field), the homology group $H_p(C,F)$ and the cohomology group $H^p(C,F)$ are indeed related by dualization: $H^p(C,F) \simeq Hom_F(H_p(C,F), F)$. 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 $F$ is a field then there is an isomorphism $H_n(C) \otimes F \simeq H_n(C \otimes F)$ and for more general $F$ 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.
Let $C_\bullet$ be a chain complex of free abelian groups. Let $A$ be an arbitrary abelian group.
Write
$C^\bullet \coloneqq Hom_{Ab}(C_\bullet, A)$ for the dual cochain complex with respect to $A$;
$H_n(C)$ for the chain homology of $C_\bullet$
$H^n(C,A)$ for the cochain cohomology of $C^\bullet$ hence for the cochain cohomology of $C_\bullet$ with coefficients in $A$.
There is a canonical morphism of abelian groups
given by sending a cocycle to evaluation of that cocycle on a chain:
(universal coefficient theorem in ordinary cohomology)
The morphism $h$ is surjective and its kernel is the Ext group $Ext^1(H_{n-1}(C, \mathbb{Z}), A)$. In other words, there is a short exact sequence
hence
Moreover, this sequence splits (non-canonically).
We reproduce the direct proof given for instance in (Boardman).
Given a homomorphism $A_1 \stackrel{f}{\to} A_2 \stackrel{g}{\to} A_3$ of abelian groups together with a retract $s : A_3 \to A_2$ of $g$, there is a short exact sequence of cokernels
Since we work in Ab, all the cokernels appearing here (as discussed there) may be expressed as quotients, e.g $coker(f) \simeq A_2/im(f)$.
The sequence of inclusions $im(g \circ f) \hookrightarrow im(g) \hookrightarrow A_3$ induces the canonical short exact sequence
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 $g$ induces a morphism $g' : A_2 / im(f) \to A_3 / im(g \circ f)$. By the existence of the retract $s$ this has itself a retract. Moreover it factors as
Therefore the first morphism here on the left has to be an isomorphism, too.
Write
for the short exact sequence of boundaries, cycles, and homology groups of $C_\bullet$ in degree $n$. Since $C_n$ is assumed to be a free abelian group and since $B_n$ and $Z_n$ 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 $H_n$. It follows that the Ext-group $Ext^1(H_n,A)$ is characterized by the short exact sequence
Notice also that the short exact sequence
is split because, as before, $B_{n-1}$ is free abelian. Using these two exact sequences on the left and right of the short exact sequence
shows that this is equivalent to
Again this splits as $B_{n-1}$ is free abelian.
In addition to these exact sequence consider the decomposition
and apply $Hom(-,A)$ to obtain the diagram
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 $i$ and $Hom(\bar \partial,A)$ are monomorphisms, it follows that the degree $n$-cocycles are
Using this for $n-1$ replaced by $n$ shows by the upper horizontal exact sequence that
Similarly the coboundaries are seen to be
Together this gives the cochain cohomology as
Now the universal coefficient theorem follows by going into lemma 1 with the identifications $A_1 = Hom(Z_{n-1}, A)$, $A_2 = Hom(B_{n-1}, A)$, $A_3 = Hom(C_n/B_n,A)$.
Let $C_\bullet \in Ch_\bullet(Ab)$ be a chain complex of free abelian groups, and let $A \in$ Ab be any abelian group. Write $C_\bullet \otimes A$ etc. for the degreewise tensor product of abelian groups.
More generally, let $R$ be a ring which is a principal ideal domain (in the above $R = \mathbb{Z}$ is the ring of integers), let $C_\bullet \in Ch_\bullet(R Mod)$ be a chain complex of free modules over $R$, let $A \in R Mod$ be any $R$-module and write $C_k \otimes_R A$ for the tensor product of modules over $R$.
For each $n \in \mathbb{N}$ there is a short exact sequence
where on the right we have the first Tor-module of the chain homology $H_{n-1}(C_\bullet)$ with $A$.
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).
For $C_\bullet$ a chain complex of free abelian groups and $A \in$ Ab any abelian group, there is a long exact sequence of the form
where $B_n$ are the boundaries and $Z_n$ the cycles of $C_\bullet$ in degree $n$ and where $i_n \colon B_n \hookrightarrow Z_n$ is the canonical inclusion.
Since, by the Dedekind-Nielsen-Schreier theorem, every subgroup of a free abelian group is itself free abelian, such as the subgroup of cycles $Z_n \hookrightarrow C_n$, it follows that for each $n \in \mathbb{N}$ we have a splitting of the short exact sequence $0 \to Z_n \to C_n \to B_{n-1}$ and hence (as discussed at split exact sequence) a direct sum decomposition
Here the second direct summand on the right identifies under the differential $\partial^C$ with the boundaries in one degree lower, since by construction $\partial^C$ is injective on $C_n/Z_n$.
Accordingly, if we regard the graded abelian groups $B_\bullet$ and $Z_\bullet$ as chain complexes with vanishing differential, then we have a sequence of chain maps
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 $(-)\otimes A$
is still a short exact sequence. The induced homology long exact sequence, as discussed there, is the long exact sequence to be shown.
of theorem 2
By lemma 2 we have short exact sequences
Since the tensor product of abelian groups is a right exact functor it preserves cokernels and hence
The dual statement were true if $(-)\otimes A$ 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
exhibits $[\cdots \to 0 \to B_n \to Z_n] \stackrel{\simeq_{qi}}{\to} H_n(C)$ as a projective resolution of $H_n(C_\bullet)$. Therefore by definition of Tor the group $Tor_1(H_n(C_\bullet), A)$ is the chain homology in degree 1 of
which is
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 $E^*$ and $F^*$ be two generalized cohomology theories and $E_*$ and $F_*$ two generalized homology theories, such that $E$ is multiplicative and $F$ is a module over $E$. Then the general problems that a Universal Coefficient Theorem should apply to are the following:
Given $E_{*}(X)$, calculate $F_{*}(X)$.
Given $E_{*}(X)$, calculate $F^{*}(X)$.
Given $E^{*}(X)$, calculate $F_{*}(X)$.
Given $E^{*}(X)$, calculate $F^{*}(X)$.
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 $E^{*}(-)$ is the generalised cohomology theory associated to a commutative ring spectrum, $E$. The cohomology theory $F^{*}(-)$ is assumed to come from a left module-spectrum over $E$, which we shall denote by $F$. We do not assume that $F$ 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.
There is a spectral sequence
with edge homomorphism
There is a spectral sequence
with edge homomorphism
For finite CW-complexes then we can derive two further statements from the above by S-duality. We use the notation $D X$ for the Spanier-Whitehead dual of $X$.
For a finite CW-complex $X$, we can apply UCT1 and UCT2 to $D X$ in place of $X$ and then use the various isomorphisms relating the cohomologies of $X$ and $D X$ to reformulate them in terms of $X$. We thus get the following statements.
For $X$ a finite CW-complex, there is a spectral sequence
with edge homomorphism
For $X$ a finite CW-complex, there is a spectral sequence
with edge homomorphism
A particularly important special case of these statements is when we have a topological space, say $Y$, and a cohomology theory, $E^{*}(-)$. Then we define a new homology theory $F_{*}(-)$ by $F_{*}(X) = E_{*}(X \wedge Y)$ and a new cohomology theory $G^{*}(-)$ by $G^{*}(X) = E^{*}(X \wedge Y)$. These are representable, the homology theory by $Y \wedge E$ and the cohomology theory by the function spectrum $F(Y,E)$. Putting these into the statements of the universal coefficient theorem, we obtain similar statements for the Künneth theorem.
There is a spectral sequence
with edge homomorphism
There is a spectral sequence
with edge homomorphism
For $X$ a finite CW-complex, there is a spectral sequence
with edge homomorphism
For $X$ a finite CW-complex, there is a spectral sequence
with edge homomorphism
The key question is, thus: when do these statements hold? Adams gives some answers in Ada69.
If $F_{*}$ is flat over $E_{*}$ then UCT1 holds, whence KT1 holds if either $E_{*}(X)$ or $E_{*}(Y)$ is flat.
If $E = S$, the sphere spectrum, then all the results are true.
If $E$ is a strict ring spectrum then KT1 holds, if also $F$ is a strict module spectrum over $E$ then UCT1 holds.
Atiyah gives a method in Ati62 for KT3 with $E =$ KU being complex K-theory and $X$, $Y$ finite complexes.
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.
(Condition 13.3 in Ada74, see also Assumption 20 in Ada69).
The spectrum $E$ is the direct limit of finite spectra $E_\alpha$ for which $E_*(D E_\alpha)$ is projective over $E_*$ and
is an isomorphism for all module-spectra $F$ over $E$. Here, $D E_\alpha$ is the S-dual of $E_\alpha$.
The main difference between the two treatments is that in Ada69, the condition involving $F$ is stated for a single module-spectrum, not for all module-spectra, and there are alternatives for homology ($F_*(-)$) and cohomology ($F^*(-)$).
In the comments following Assumption 20 in Ada69, Adams remarks that this is implied by a stronger condition (Proposition 17) on $E$ which makes no reference to $F$. As $E$ is a ring spectrum, this reduces to:
With this assumption, Adams shows the following result:
Let $E$ be a ring spectrum satisfying the assumption above. Let $F$ be a module-spectrum over $E$. Then 2 holds, and the spectral sequence is convergent.
What “convergent” means here is spelled out in Ada74, Theorem 8.2.
Let $E$ be a ring spectrum satisfying the assumption above. Suppose that $E_*(X)$ is projective over $E_*$. Then the spectral sequence from 2 collapses at the $E^2$ term. That is,
is an isomorphism.
In Ada74, Adams lists several cohomology theories (for $E^*(-)$) where the assumption holds. These are: $S$, $H\mathbb{Z}_p$, $MO$, $MU$, $MSp$, $K$, $KO$.
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, $E^*(-)$ is a multiplicative generalised cohomology theory with representing ring spectrum $E$. We use $E_*(-)$ for the associated homology theory. Following Boa95 and BJW95, cohomology and homology are not reduced in this section.
Assume that $E_*(X)$ or $E_*(Y)$ is a free or flat $E^*$-module. Then the pairing:
induces the Künneth isomorphism $E_*(X \times Y) \cong E_*(X) \otimes E_*(Y)$ in homology.
The next result relates homology and cohomology.
Assume that $E_*(X)$ is a free $E^*$-module. Then $X$ has strong duality, i.e. the duality map $d \colon E^*(X) \to D E_*(X)$ is a homeomorphism between the profinite topology on $E^*(X)$ and the dual-finite topology on $D E_*(X)$. In particular, $E^*(X)$ is complete Hausdorff.
Combining these two gives the Künneth theorem for cohomology.
Assume that $E_*(X)$ and $E_*(Y)$ are free $E^*$-modules. Then we have the Künneth homeomorphism $E^*(X \times Y) \cong E^*(X) \widehat{\otimes} E^*(Y)$ in cohomology.
There are similar results for spectra. Boardman, Johnson, and Wilson write reduced homology and cohomology as $E_*(X,o)$ and $E^*(X,o)$, even when $X$ is a spectrum (and so the reduced theories are all that there are).
Assume that $E_*(X,o)$ or $E_*(Y,o)$ is a free or flat $E^*$-module. Then the pairing $\times \colon E_*(X,o) \otimes E_*(Y,o) \to E_*(X \wedge Y,o)$ is an isomorphism in homology.
Assume that $E_*(X,o)$ is a free $E^*$-module. Then $X$ has strong duality, i.e. $d \colon E^*(X,o) \to D E_*(X,o)$ is a homeomorphism between the profinite topology on $E^*(X,o)$ and the dual-finite topology on $D E_*(X,o)$. In particular, $E^*(X,o)$ is complete Hausdorff.
Assume that $E_*(X,o)$ and $E_*(Y,o)$ are free $E^*$-modules. Then the pairing
induces the cohomology Künneth homeomorphism.
For $X$ a topological space, write $Sing X$ for its singular simplicial complex and
for the normalized chain complex of the simplicial abelian group obtained by forming degreewise the free abelian group.
The singular homology $H_\bullet(X)$ of $X$ is the chain homology of $C_\bullet(X)$, and for $A$ some coefficient abelian group, the singular cohomology $H^\bullet(X,A)$ is the cochain cohomology, of $C_\bullet(X)$ with coefficients in $A$.
Comparison with the ordinary universal coefficient theorem 1 shows that:
(universal coefficient theorem in topology)
For $X$ a topological space, $A$ an abelian group and $n \geq 1 \in \mathbb{N}$, the singular homology and singular cohomology of $X$ fit into a split short exact sequence of the form
An exposition of the universal coefficient theorem for ordinary cohomology and homology is in section 3.1 of
The note
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
Universal coefficient theorems for generalized homology are discussed in
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.
Discussion for KK-theory is in