universal coefficient theorem (changes) in nLab

Proof (of theorem 1)

Write

0→B n→Z n→H n→0 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)0→Z n→C n→∂B n−1→0 0 \to Z_n \to C_n \stackrel{\partial}{\to} B_{n-1} \to 0

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

0→Z n/B n→C n/B n→C n/Z n→0 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)0→H n→C n/B n→∂B n−1. 0 \to H_n \to C_n/B_n \stackrel{\partial}{\to} B_{n-1} \,.

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

In addition to these exact sequence consider the decomposition

∂:C n→C n/B n→C n/Z n→≃B n−1↪Z n−1↪C n−1 \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 n−1,A) ← Hom(Z n−1,A) ↑ ↖ ↑ 0 Hom(C n−1,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 n−1≔ker(Hom(C n−1,A)→Hom(C n,a))≃ker(Hom(C n−1,A)→Hom(B n−1,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 n−1n-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 n≔imHom(∂,A)≃im(Hom(Z n−1,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 n≃coker(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 n−1,A)A_1 = Hom(Z_{n-1}, A), A 2=Hom(B n−1,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 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 (for use in the Adams spectral sequence, see there for more) 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, III.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.

\begin{statement} (UCT1) \label{ucta}

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).

\end{statement}

\begin{statement} (UCT2) \label{uctb}

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^{*}).

\end{statement}

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.

\begin{statement} (UCT3) \label{uctc}

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).

\end{statement}

\begin{statement} (UCT4) \label{utcd}

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_{*}).

\end{statement}

(This is a generalization of the Kronecker pairing, see also e.g. Schwede 12, prop. 6.20).

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 *(X∧Y)F_{*}(X) = E_{*}(X \wedge Y) and a new cohomology theory G *(−)G^{*}(-) by G *(X)=E *(X∧Y)G^{*}(X) = E^{*}(X \wedge Y). These are representable, the homology theory by Y∧EY \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.

\begin{statement} (KT1) \label{kta}

There is a spectral sequence

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

with edge homomorphism

E *(X)⊗ E *E *(Y)→E *(X∧Y). E_{*}(X) \otimes _{E_{*}} E_{*}(Y) \to E_{*}(X \wedge Y).

\end{statement}

\begin{statement} (KT2) \label{ktb}

There is a spectral sequence

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

with edge homomorphism

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

\end{statement}

\begin{statement} (KT3) \label{ktc}

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

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

with edge homomorphism

E *(X)⊗ E *E *(Y)→E *(X∧Y). E^{*}(X) \otimes _{E^{*}} E^{*}(Y) \to E^{*}(X \wedge Y).

\end{statement}

\begin{statement} (KT4) \label{ktd}

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

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

with edge homomorphism

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

\end{statement}

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

• If F *F_{*} is flat over E *E_{*} then UCT1 holds, whence KT1 holds if either E *(X)E_{*}(X) or E *(Y)E_{*}(Y) is flat.

• If E=SE = S, the sphere spectrum, then all the results are true.

• If EE is a strict ring spectrum then KT1 holds, if also FF is a strict module spectrum over EE then UCT1 holds.

• Atiyah gives a method in Ati62 for KT3 with E=E = KU being complex K-theory and XX, YY finite complexes.

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.

\begin{assum} (Condition 13.3 in Ada74, see also Assumption 20 in Ada69) \label{adams_assumption}

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.

\end{assum}

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:

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

\begin{corollary} \label{uctproj} Let EE be a ring spectrum satisfying Assumption \ref{adams_assumption}. Suppose that E *(X)E_*(X) is projective over E *E_*. Then the spectral sequence from \ref{uctb} 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. \end{corollary}

In Ada74, Adams lists several cohomology theories (for E *(−)E^*(-)) where the assumption holds. These are: SS, Hℤ 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.

The next result relates homology and cohomology.

Combining these two gives the Künneth theorem for 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).

Examples

For singular cohomology

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

C •(X)≔Nℤ[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:

(e.g. Moerman 15, Cor. 1.2.1)

• Künneth theorem

• Eilenberg-Zilber theorem

• bootstrap category

References

For ordinary (co)homology

• Edwin Spanier, section 5.5 of Algebraic topology, 1966

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

• Allen Hatcher, Algebraic topology (pdf); also section 3.A.

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

• Michael Boardman, The universal coefficient theorem (pdf)

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

The universal coefficient theorem in symmetric monoidal model categories of spectra is discussed in

• Anthony Elmendorf, Igor Kriz, Peter May, section 8 of Modern foundations for stable homotopy theory (pdf)

Universal coefficient theorems for generalized homology are discussed in in:

• Friedrich Bauer, Remarks on universal coefficient theorems for generalized homology theories Quaestiones Mathematicae Volume 9, Issue 1 & 4, 1986, Pages 29 - 549 1, 4 (1986) 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.Stable Operations in Generalized Cohomology [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

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

See also

• Stefan Schwede, prop. 6.20 of Symmetric spectra, 2012 (pdf)

Further discussion along these lines includes

• Andrew Baker, Andrey Lazarev, On the Adams Spectral Sequence for RR-modules, Algebr. Geom. Topol. 1 (2001) 173-199 (arXiv.0105079)

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)