Editing Universal coefficient theorem

{{short description|Establish relationships between homology and cohomology theories}}
In [[algebraic topology]], '''universal coefficient theorems'''   establish relationships between [[homology group]]s (or [[cohomology group]]s) with different coefficients. For instance, for every [[topological space]] {{mvar|X}}, its ''integral homology groups'':

:<math>H_i(X,\Z)</math>

completely determine its ''homology groups with coefficients in'' {{mvar|A}}, for any [[abelian group]] {{mvar|A}}:

:<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 complex]]es of [[free abelian group]]s. The form of the result is that other coefficients {{mvar|A}} 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 (algebra)|torsion]] in the homology. Quite generally, the result indicates the relationship that holds between the [[Betti number]]s <math>b_i</math> of <math>X</math> and the Betti numbers <math>b_{i,F}</math> with coefficients in a [[field (mathematics)|field]] <math>F</math>. These can differ, but only when the [[characteristic (algebra)|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 case ==
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 [[splitting lemma|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>.
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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 cohomology==
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 functor|adjoint]]'' to the homology [[functor]].<ref>{{Harv|Kainen|1971}}</ref>

== Example: mod 2 cohomology of the real projective space==
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>

==Corollaries==
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 [[Fundamental theorem of finitely generated abelian groups#Classification|decomposition]].

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

where <math>\beta_i(X)</math> are the [[Betti number]]s 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 [[orientability|orientable]], [[closed manifold|closed]], and [[connected space|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 sequence ==

There is a generalization of the universal coefficient theorem for (co)homology with [[Twisted Poincaré duality|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 functor|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 functor|Tor group]] and the differential <math>d_r</math> having degree <math>(r-1,-r)</math>.

== Notes ==
{{reflist}}

==References==
*[[Allen Hatcher]], ''Algebraic Topology'', Cambridge University Press, Cambridge, 2002. {{ISBN|0-521-79540-0}}. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html author's homepage].
* {{cite journal
  | last = Kainen
  | first = P. C.
  | authorlink = Paul Chester Kainen
  | title = Weak Adjoint Functors
  | journal = Mathematische Zeitschrift 
  | volume = 122
  | issue = 
  | pages = 1–9
  | year = 1971
  | pmid =   
  | pmc = 
  | doi = 10.1007/bf01113560
  | s2cid = 122894881
 }}
*[[Jerome Levine]]. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

== External links ==
*[https://math.stackexchange.com/q/768481 Universal coefficient theorem with ring coefficients]

[[Category:Homological algebra]]
[[Category:Theorems in algebraic topology]]