Universal coefficient set

The universal coefficient theorem is a statement of a more technical character from the mathematical subfield of algebraic topology . It allows the homology or cohomology of a space with coefficients in any Abelian group to be calculated from the homology or cohomology with coefficients in whole numbers .

Homological version

Let it be a topological space , an Abelian group and a natural number. Then there is a natural short exact sequence ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle n}$

${\displaystyle 0\to H_{n}(X)\otimes A\to H_{n}(X;A)\to \operatorname {Tor} _{1}^{\mathbb {Z} }(H_{n-1}(X),A)\to 0.}$

The abbreviation stands for , and Tor is the torsion product . ${\displaystyle H_{n}(X)}$ ${\displaystyle H_{n}(X;\mathbb {Z} )}$

The episode divides , but not naturally.

Cohomological version

Let it be a topological space , an Abelian group and a natural number. Then there is a natural short exact sequence ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle n}$

${\displaystyle 0\to \operatorname {Ext} _{\mathbb {Z} }^{1}(H_{n-1}(X),A)\to H^{n}(X;A)\to \operatorname {Hom} (H_{n}(X),A)\to 0.}$

Again stands for , and Ext is the derived functor Ext . The homomorphism is defined by the Kronecker pairing . ${\displaystyle H_{n}(X)}$ ${\displaystyle H_{n}(X;\mathbb {Z} )}$ ${\displaystyle H^{n}(X;A)\to \operatorname {Hom} (H_{n}(X),A)}$

In contrast to the homological version, this statement itself is not trivial. ${\displaystyle A=\mathbb {Z} }$

As above, the sequence divides, but not naturally.

Application examples

• Along with the statement follows ${\displaystyle H_{1}(X)=\pi _{1}(X)^{\mathrm {ab} }}$

${\displaystyle H^{1}(X;\mathbb {R} )=\operatorname {Hom} (\pi _{1}(X),\mathbb {R} ).}$

• The real projective plane has the 2-sphere as a two-leaf, universal superposition , so it is true that one has to ${\displaystyle X=\mathbb {R} P^{2}}$ ${\displaystyle H_{1}(X)=\pi _{1}(X)=\mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle H^{2}(X;A)}$

${\displaystyle \operatorname {Ext} ^{1}(\mathbb {Z} /2\mathbb {Z} ,A)=A/2A}$

isomorphic subgroup.

Generalizations

• There are completely analogous statements for any flat (for homology) or free (for cohomology) chain complexes over an arbitrary principal ideal and - modules . ${\displaystyle R}$ ${\displaystyle R}$
• The set of Künneth contains the universal Koeffiziententheorem as a special case.

swell

• JP May, A Concise Course in Algebraic Topology . University of Chicago Press, Chicago 1999. ISBN 0-226-51183-9 , chapter 17.