Universal coefficient theorem

Statement of the homology case

Consider the tensor product of modules H_i(X,\Z)\otimes A. The theorem states there is a short exact sequence involving the Tor functor

: 0 \to H_i(X, \Z)\otimes A , \overset{\mu}\to , H_i(X,A) \to \operatorname{Tor}1(H{i-1}(X, \Z),A)\to 0.

Furthermore, this sequence splits, though not naturally. Here \mu is the map induced by the bilinear map H_i(X,\Z)\times A\to H_i(X,A).

If the coefficient ring A is \Z/p\Z, this is a special case of the Bockstein spectral sequence.

Universal coefficient theorem for cohomology

Let G be a module over a principal ideal domain R (for example \Z, or any field.)

There is a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence

: 0 \to \operatorname{Ext}R^1(H{i-1}(X; R), G) \to H^i(X; G) , \overset{h} \to , \operatorname{Hom}_R(H_i(X; R), G)\to 0.

As in the homology case, the sequence splits, though not naturally. In fact, suppose

:H_i(X;G) = \ker \partial_i \otimes G / \operatorname{im}\partial_{i+1} \otimes G,

and define

:H^*(X; G) = \ker(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G)).

Then h above is the canonical map:

:h([f])([x]) = f(x).

An alternative point of view can be based on representing cohomology via Eilenberg–MacLane space, where the map h takes a homotopy class of maps X\to K(G,i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.

Example: mod 2 cohomology of the real projective space

Let X=\mathbb{RP}^n, the real projective space. We compute the singular cohomology of X with coefficients in G=\Z/2\Z using integral homology, i.e., R=\Z.

Knowing that the integer homology is given by:

:H_i(X; \Z) = \begin{cases} \Z & i = 0 \text{ or } i = n \text{ odd,}\ \Z/2\Z & 0 0 & \text{otherwise.} \end{cases}

We have \operatorname{Ext}(G,G)=G and \operatorname{Ext}(R,G)=0, so that the above exact sequences yield :H^i (X; G) = G for all i=0,\dots,n. In fact the total cohomology ring structure is

:H^*(X; G) = G [w] / \left \langle w^{n+1} \right \rangle.

Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex X, H_i(X,\Z) is finitely generated, and so we have the following decomposition.

: H_i(X; \Z) \cong \Z^{\beta_i(X)}\oplus T_{i},

where \beta_i(X) are the Betti numbers of X and T_i is the torsion part of H_i. One may check that

: \operatorname{Hom}(H_i(X),\Z) \cong \operatorname{Hom}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Hom}(T_i, \Z) \cong \Z^{\beta_i(X)},

and

:\operatorname{Ext}(H_i(X),\Z) \cong \operatorname{Ext}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Ext}(T_i, \Z) \cong T_i.

This gives the following statement for integral cohomology:

: H^i(X;\Z) \cong \Z^{\beta_i(X)} \oplus T_{i-1}.

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that \beta_i(X)=\beta_{n-i}(X).

Universal coefficient spectral sequence

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.

For cohomology we have

:E^{p,q}2=\operatorname{Ext}{R}^q(H_p(C_),G)\Rightarrow H^{p+q}(C_;G),

where R is a ring with unit, C_* is a chain complex of free modules over R, G is any (R,S)-bimodule for some ring with a unit S, and \operatorname{Ext} is the Ext group. The differential d^r has degree (1-r,r).

Similarly for homology, :E_{p,q}^2=\operatorname{Tor}^{R}q(H_p(C),G)\Rightarrow H_(C_*;G), for \operatorname{Tor} the Tor group and the differential d_r having degree (r-1,-r).

Notes

References

• Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. . A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
• {{cite journal
• Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

References

1. {{Harv. Kainen. 1971