Universal coefficient theorem

The universal Koeffiziententheorem is a statement rather technical nature of the mathematical subfield of algebraic topology. It allows to calculate the homology and cohomology of a space with coefficients in an arbitrary abelian group of the homology and cohomology with coefficients in the integers.

Homological version

Let be a topological space, an abelian group and a natural number. Then there is a natural short exact sequence

It is an abbreviation for, and port is the Torsionsprodukt.

The sequence splits, but not naturally.

Cohomological version

Let be a topological space, an abelian group and a natural number. Then there is a natural short exact sequence

The focus is again short for, and Ext is the derived functor Ext

In contrast to the homological version of this statement is not trivial even for.

Splits above does not, of course, the result, however.

Application Examples

  • Follows together with the statement
  • The real projective plane, the 2-sphere as a two-bladed, universal cover, so true, so has to

Generalizations

  • There are completely analogous statements for any flat ( for homology ) or free ( for cohomology ) chain complexes over an arbitrary principal ideal ring and modules.
  • The set of Kiinneth 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
  • Algebraic Topology
  • Homological Algebra
  • Set ( mathematics)
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