Cap product

In algebraic topology , a branch of mathematics , the cap product defines a link between the cohomology and homology of a space .

definition

Let be a topological space , let the -th singular chain group, i.e. the free Abelian group over the set of all continuous maps of the standard simplex after and . Use or to denote the inclusions of the standard or simplex as the “front- dimensional side” or “rear- dimensional side” in the standard simplex. ${\displaystyle X}$ ${\displaystyle C_{n}(X)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \Delta ^{n}}$ ${\displaystyle X}$ ${\displaystyle C^{n}(X)=Hom(C_{n}(X),\mathbb {Z} )}$ ${\displaystyle \iota _{0\ldots p}\colon \Delta ^{p}\rightarrow \Delta ^{p+q}}$ ${\displaystyle \iota _{p\ldots p+q}\colon \Delta ^{q}\rightarrow \Delta ^{p+q}}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle (p+q)}$

For and a singular simplex (with ) is defined ${\displaystyle \psi \in C^{q}(X)}$ ${\displaystyle \sigma :\Delta ^{p}\rightarrow X}$ ${\displaystyle p\geq q}$

${\displaystyle \sigma \frown \psi :=(-1)^{pq}\psi (\sigma \circ \iota _{0\ldots q})\sigma \circ \iota _{q\ldots p}}$

and sets this linearly to a mapping

${\displaystyle C^{q}(X)\times C_{p}(X)\rightarrow C_{p-q}(X)}$

away.

More generally be a ring and be . Then you get a picture ${\displaystyle R}$ ${\displaystyle C_{n}(X;R)=C_{n}(X)\otimes _{\mathbb {Z} }R,C^{n}(X)=Hom(C_{n}(X),R)}$

${\displaystyle C^{q}(X;R)\times C_{p}(X;R)\rightarrow C_{p-q}(X;R)}$.

From the relation

${\displaystyle \partial (\sigma \frown \psi )=(-1)^{q}(\partial \sigma \frown \psi -\sigma \frown \delta \psi )}$

it follows that the cap product is a well-defined map

${\displaystyle H^{q}(X;R)\times H_{p}(X;R)\rightarrow H_{p-q}(X;R)}$

Are defined.

properties

The following applies to continuous images ${\displaystyle f:X\rightarrow Y}$

${\displaystyle f_{*}(c)\frown \psi =f_{*}(c\frown f^{*}(\psi ))}$

with , . ${\displaystyle c\in C_{p}(X;R)}$ ${\displaystyle \psi \in C^{q}(Y;R)}$

The cap product is related to the cup product by the following equation:

${\displaystyle \psi (c\frown \varphi )=(\varphi \smile \psi )(c)}$

for , , ${\displaystyle c\in C_{p}(X;R)}$ ${\displaystyle \psi \in C^{q}(X;R)}$ ${\displaystyle \varphi \in C^{p-q}(X;R).}$

Application: Poincaré duality

→ Main article : Poincaré duality

Be a closed, orientable -manifold and ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle \left[M\right]\in H_{n}(M;\mathbb {Z} )}$

the fundamental class . Then the Cap product realizes with an isomorphism ${\displaystyle \left[M\right]}$

${\displaystyle H^{k}(M;\mathbb {Z} )\rightarrow H_{n-k}(M;\mathbb {Z} )}$

for . ${\displaystyle k=0,\ldots ,n}$

literature

• Glen Bredon : Topology and Geometry. Corrected third printing of the 1993 original. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1997. ISBN 0-387-97926-3
• Allen Hatcher : Algebraic Topology , Cambridge University Press (2002) ISBN 0-521-79540-0 .
• Ralph Stöcker, Heiner Zieschang : Algebraic Topology. An introduction. Second edition. Math guides. BG Teubner, Stuttgart, 1994. ISBN 3-519-12226-X