Template:Short description In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space Template:Mvar, its integral homology groups:
- <math>H_i(X,\Z)</math>
completely determine its homology groups with coefficients in Template:Mvar, for any abelian group Template:Mvar:
- <math>H_i(X,A)</math>
Here <math>H_i</math> might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients Template:Mvar may be used, at the cost of using a Tor functor.
For example, it is common to take <math>A</math> to be <math>\Z/2\Z</math>, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers <math>b_i</math> of <math>X</math> and the Betti numbers <math>b_{i,F}</math> with coefficients in a field <math>F</math>. These can differ, but only when the characteristic of <math>F</math> is a prime number <math>p</math> for which there is some <math>p</math>-torsion in the homology.
Statement of the homology caseEdit
Consider the tensor product of modules <math>H_i(X,\Z)\otimes A</math>. The theorem states there is a short exact sequence involving the Tor functor
- <math> 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.</math>
Furthermore, this sequence splits, though not naturally. Here <math>\mu</math> is the map induced by the bilinear map <math>H_i(X,\Z)\times A\to H_i(X,A)</math>.
If the coefficient ring <math>A</math> is <math>\Z/p\Z</math>, this is a special case of the Bockstein spectral sequence.
Universal coefficient theorem for cohomologyEdit
Let <math>G</math> be a module over a principal ideal domain <math>R</math> (for example <math>\Z</math>, 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
- <math> 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.</math>
As in the homology case, the sequence splits, though not naturally. In fact, suppose
- <math>H_i(X;G) = \ker \partial_i \otimes G / \operatorname{im}\partial_{i+1} \otimes G,</math>
and define
- <math>H^*(X; G) = \ker(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G)).</math>
Then <math>h</math> above is the canonical map:
- <math>h([f])([x]) = f(x).</math>
An alternative point of view can be based on representing cohomology via Eilenberg–MacLane space, where the map <math>h</math> takes a homotopy class of maps <math>X\to K(G,i)</math> to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.<ref>Template:Harv</ref>
Example: mod 2 cohomology of the real projective spaceEdit
Let <math>X=\mathbb{RP}^n</math>, the real projective space. We compute the singular cohomology of <math>X</math> with coefficients in <math>G=\Z/2\Z</math> using integral homology, i.e., <math>R=\Z</math>.
Knowing that the integer homology is given by:
- <math>H_i(X; \Z) =
\begin{cases} \Z & i = 0 \text{ or } i = n \text{ odd,}\\ \Z/2\Z & 0<i<n,\ i\ \text{odd,}\\ 0 & \text{otherwise.} \end{cases}</math>
We have <math>\operatorname{Ext}(G,G)=G</math> and <math>\operatorname{Ext}(R,G)=0</math>, so that the above exact sequences yield
- <math>H^i (X; G) = G</math>
for all <math>i=0,\dots,n</math>. In fact the total cohomology ring structure is
- <math>H^*(X; G) = G [w] / \left \langle w^{n+1} \right \rangle.</math>
CorollariesEdit
A special case of the theorem is computing integral cohomology. For a finite CW complex <math>X</math>, <math>H_i(X,\Z)</math> is finitely generated, and so we have the following decomposition.
- <math> H_i(X; \Z) \cong \Z^{\beta_i(X)}\oplus T_{i},</math>
where <math>\beta_i(X)</math> are the Betti numbers of <math>X</math> and <math>T_i</math> is the torsion part of <math>H_i</math>. One may check that
- <math> \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)},</math>
and
- <math>\operatorname{Ext}(H_i(X),\Z) \cong \operatorname{Ext}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Ext}(T_i, \Z) \cong T_i.</math>
This gives the following statement for integral cohomology:
- <math> H^i(X;\Z) \cong \Z^{\beta_i(X)} \oplus T_{i-1}. </math>
For <math>X</math> an orientable, closed, and connected <math>n</math>-manifold, this corollary coupled with Poincaré duality gives that <math>\beta_i(X)=\beta_{n-i}(X)</math>.
Universal coefficient spectral sequenceEdit
There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.
For cohomology we have
- <math>E^{p,q}_2=\operatorname{Ext}_{R}^q(H_p(C_*),G)\Rightarrow H^{p+q}(C_*;G),</math>
where <math>R</math> is a ring with unit, <math>C_*</math> is a chain complex of free modules over <math>R</math>, <math>G</math> is any <math>(R,S)</math>-bimodule for some ring with a unit <math>S</math>, and <math>\operatorname{Ext}</math> is the Ext group. The differential <math>d^r</math> has degree <math>(1-r,r)</math>.
Similarly for homology,
- <math>E_{p,q}^2=\operatorname{Tor}^{R}_q(H_p(C_*),G)\Rightarrow H_*(C_*;G),</math>
for <math>\operatorname{Tor}</math> the Tor group and the differential <math>d_r</math> having degree <math>(r-1,-r)</math>.
NotesEdit
ReferencesEdit
- Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Template:ISBN. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
- Template:Cite journal
- Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498