Contents

cohomology

# Contents

## Idea

For $X \overset{f}{\longrightarrow} Y$ a continuous function, $u,v \in H^{2k}(Y)$ two cocycles (in ordinary cohomology) of even degree, and $a,b$ cochains on $X$ witnessing trivializations of the pullback $f^\ast u$ and of the cup product $u \cup v$, respectively, the cocycle expression

$a \cup f^\ast v - f^\ast b \;\in\; H^{4k-1}(X)$

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 November 26, 2020 at 16:13:33. See the history of this page for a list of all contributions to it.