Cup product

In algebraic topology, the cup product denotes a multiplicative structure on a cohomology . This gives a ring structure on the cohomology , which is referred to as a cohomology ring . There is no analogous product for homologies .

The cup product defines a product for topological spaces and natural numbers ${\displaystyle X}$ ${\displaystyle p,q}$

${\displaystyle H^{p}(X)\times H^{q}(X)\rightarrow H^{p+q}(X)}$

${\displaystyle (\alpha ,\beta )\rightarrow \alpha \cup \beta }$

with the properties

${\displaystyle \alpha \cup \beta =(-1)^{pq}\beta \cup \alpha }$(graduated commutativity )

${\displaystyle f^{*}(\alpha \cup \beta )=f^{*}\alpha \cup f^{*}\beta }$for all continuous images (naturalness) ${\displaystyle f\colon Y\rightarrow X}$

${\displaystyle \alpha \cup (\beta _{1}+\beta _{2})=\alpha \cup \beta _{1}+\alpha \cup \beta _{2}}$( Distributivity )

${\displaystyle \alpha \cup (\beta \cup \gamma )=(\alpha \cup \beta )\cup \gamma }$( Associativity ).

definition

Below are three definitions for the cup product. The definition of the cup product for singular cohomology is the most general of the three and includes the definitions for de Rham and simplicial cohomology.

De Rham cohomology

This definition assumes that there is a differentiable manifold . ${\displaystyle X}$

In De Rham cohomology , cohomology classes are represented by differential forms . Leibniz's rule applies to the outer product of differential forms . It may therefore be the cup-product of the of and represented Kohomologieklassen by ${\displaystyle \omega \in \Omega ^{p}(X),\eta \in \Omega ^{q}(X)}$ ${\displaystyle d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{p}\omega \wedge d\eta }$ ${\displaystyle \omega }$ ${\displaystyle \eta }$ ${\displaystyle \alpha =\left[\omega \right],\beta =\left[\eta \right]}$

${\displaystyle \left[\omega \right]\cup \left[\eta \right]=\left[\omega \wedge \eta \right]}$

define and, because of the Leibniz rule, receive a well-defined mapping of the cohomology groups.

Simplicial cohomology

This definition assumes that it is a simplicial complex . ${\displaystyle X}$

In simplicial cohomology , cohomology classes are represented by homomorphisms , where the -th chain group , i.e. the free Abelian group, is above the set of -simplices of the simplicial complex . For a simplex , we denote by or the sub-implices spanned by the first and last corners. For two homomorphisms , one defines by ${\displaystyle \alpha \in H^{n}(X;R)}$ ${\displaystyle f\colon C_{n}(X)\rightarrow R}$ ${\displaystyle C_{n}(X)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle (p+q)}$ ${\displaystyle \left[v_{0},\ldots ,v_{p+q}\right]}$ ${\displaystyle \left[v_{0},\ldots ,v_{p}\right]}$ ${\displaystyle \left[v_{p},\ldots ,v_{p+q}\right]}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle f\colon C_{p}(X)\rightarrow R}$ ${\displaystyle g\colon C_{q}(X)\rightarrow R}$ ${\displaystyle f\cup g:C_{p+q}(X)\rightarrow R}$

${\displaystyle (f\cup g)(\left[v_{0},\ldots ,v_{n}\right])=f(\left[v_{0},\ldots ,v_{p}\right])g(\left[v_{p},\ldots ,v_{p+q}\right])}$.

This link fulfills the Leibniz rule , so a well-defined mapping of the cohomology groups is obtained by defining the cup product of the cohomology classes of and as the cohomology class of . ${\displaystyle d(f\cup g)=df\cup g+(-1)^{p}f\cup dg}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f\cup g}$

Singular cohomology

This definition works for any topological spaces; in the case of differentiable manifolds or simplicial complexes, the ring structure defined in this way on the singular cohomology is isomorphic to the ring structures defined above on De-Rham or simplicial cohomology.

Let be a ring and the singular cohomology with coefficients in . Cohomology classes are represented by homomorphisms , whereby the -th singular chain group, i.e. the free Abelian group, is over the set of all continuous mappings of the standard simplex after . With one calls or the inclusions of the standard - or -Simplexes as "front -dimensional side" or "rear -dimensional page" in the default simplex. For a singular simplex and coquette , one defines ${\displaystyle R}$ ${\displaystyle H^{\bullet }(X;R)}$ ${\displaystyle R}$ ${\displaystyle \alpha \in H^{n}(X;R)}$ ${\displaystyle f\colon C_{n}(X)\rightarrow R}$ ${\displaystyle C_{n}(X)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \Delta ^{n}}$ ${\displaystyle X}$ ${\displaystyle \iota _{0\ldots p}\colon \Delta ^{p}\rightarrow \Delta ^{p+q}}$ ${\displaystyle \iota _{p\ldots p+q}\colon \Delta ^{q}\rightarrow \Delta ^{p+q}}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle (p+q)}$ ${\displaystyle (p+q)}$ ${\displaystyle \sigma :\Delta ^{p+q}\rightarrow X}$ ${\displaystyle f\colon C_{p}(X)\rightarrow R}$ ${\displaystyle g\colon C_{q}(X)\rightarrow R}$

${\displaystyle (f\cup g)(\sigma )=f(\sigma \circ \iota _{0\ldots p})g(\sigma \circ \iota _{p\ldots p+q})}$.

This link fulfills the Leibniz rule , so a well-defined mapping of the cohomology groups is obtained by defining the cup product of the cohomology classes of and as the cohomology class of . ${\displaystyle d(f\cup g)=df\cup g+(-1)^{p}f\cup dg}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f\cup g}$

The cup product defines an additional, multiplicative structure on the cohomology groups. With the help of this multiplicative structure it is sometimes possible to distinguish spaces whose cohomology groups are isomorphic as (additive) Abelian groups.

Cut and signature

For a closed , orientable -dimensional manifold there is an isomorphism . The cup product thus defines a symmetrical bilinear shape ${\displaystyle 4n}$ ${\displaystyle M}$ ${\displaystyle H^{4n}(M;\mathbb {Z} )\cong \mathbb {Z} }$

${\displaystyle H^{2n}(M;\mathbb {Z} )\times H^{2n}(M;\mathbb {Z} )\rightarrow \mathbb {Z} }$,

the so-called cut shape.

The signature of is by definition the signature of this symmetrical bilinear form. The Hirzebruch signature theorem states that one signature as a polynomial in the Pontryagin classes can represent. ${\displaystyle M}$

Simply connected differentiable 4-manifolds are classified by their intersection form, except for homeomorphism (but not diffeomorphism). For the classification of simply connected topological 4-manifolds, one needs the Kirby-Siebenmann invariant in addition to the sectional form.

literature

• A. Hatcher: Algebraic Topology. Cambridge Univ. Press, Cambridge 2010, ISBN 978-0-521-79160-1 . Chapter 3.2 (on: math.cornell.edu , PDF; 539 kB).

supporting documents

1. ↑ H. Weyl: Analisis situs combinatorio. In: Revista Matematica HispanoAmericana. 5, 1923, pp. 390-432.
2. ^ Friedrich Hirzebruch : New topological methods in algebraic geometry. In: Results of mathematics and its border areas. (NF), Issue 9. Springer-Verlag, Berlin / Göttingen / Heidelberg 1956, Chapter 2.
3. ↑ Michael Freedman : The topology of four-dimensional manifolds. In: J. Differential Geom. 17, no. 3, 1982, pp. 357-453.