Kronecker mating

In the mathematical field of algebraic topology , the Kronecker pairing defines a pairing between homology and cohomology .

definition

Let it be a topological space, a natural number , a homology class and a cohomology class with coefficients in an Abelian group . Then the Kronecker pairing is from and through ${\displaystyle X}$ ${\displaystyle k}$ ${\displaystyle h\in H_{k}(X;\mathbb {Z} )}$ ${\displaystyle \beta \in H^{k}(X;A)}$ ${\displaystyle A}$ ${\displaystyle \beta }$ ${\displaystyle h}$

${\displaystyle \langle \beta ,h\rangle :=c(z)\in A}$

defined, wherein a the cohomology class representing cocycle and a homology class representing Cycles is. ${\displaystyle c\in C^{k}(X;A)}$ ${\displaystyle \beta }$ ${\displaystyle z\in C_{k}(X)}$ ${\displaystyle h}$

It can be shown that the Kronecker pairing is well-defined , i.e. that the value of does not depend on the selection of the cocycle representing the cohomology class or the cycle representing the homology class . ${\displaystyle c(z)}$ ${\displaystyle c}$ ${\displaystyle z}$

Surjectivity

From the universal coefficient theorem it follows that the homomorphism defined by the Kronecker pairing

${\displaystyle H^{k}(X;A)\to \operatorname {Hom} (H_{k}(X),A)}$

is an epimorphism .

literature

• Ralph Stöcker, Heiner Zieschang : Algebraic Topology. An introduction. Second edition. Math guides. BG Teubner, Stuttgart, 1994. ISBN 3-519-12226-X .

Web links

• Kronecker pairing (nLab)