nLab
functional cup product

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For XfYX \overset{f}{\longrightarrow} Y a continuous function, u,vH 2k(Y)u,v \in H^{2k}(Y) two cocycles in even degree, and a,ba,b cochains on XX witnessing trivializations of the pullback f *uf^\ast u and of the cup product uvu \cup v, respectively, the cocycle expression

af *vf *bH 4k1(X) a \cup f^\ast v - f^\ast b \;\in\; H^{4k-1}(X)

is called a functional cup product in (Steenrod 49).

This appears notably as the homotopy version of Whitehead's integral formula (Whitehead 47) for the Hopf invariant (see Haefliger 78, Remark on p. 17, Griffith-Morgan 81, Section 14.5). More recently Sinha-Walter 13, Example 1.9 speak of homotopy period expressions. A transparent proof is given in FSS 19, relating to the Hopf-Wess-Zumino term of the M5-brane.

References

Last revised on October 2, 2019 at 14:19:28. See the history of this page for a list of all contributions to it.