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
The abbreviation stands for , and Tor is the torsion product .
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
Again stands for , and Ext is the derived functor Ext . The homomorphism is defined by the Kronecker pairing .
In contrast to the homological version, this statement itself is not trivial.
As above, the sequence divides, but not naturally.
Application examples
- Along with the statement follows
- The real projective plane has the 2-sphere as a two-leaf, universal superposition , so it is true that one has to
- 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 .
- 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.