nLab functional cup product

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.

Related concepts

• homotopy Whitehead integral formula

• intersection pairing

• Hopf-Wess-Zumino term

References

• J. H. C. Whitehead, An expression of Hopf’s invariant as an integral, Proc. Nat. Acad. Sci. USA 33 (1947), 117–123 (jstor:87688)

• Norman Steenrod, Cohomology Invariants of Mappings, Annals of Mathematics Second Series, Vol. 50, No. 4 (Oct., 1949), pp. 954-988 (jstor:1969589)

• André Haefliger, p. 3 of Whitehead products and differential forms, In: P.A. Schweitzer (ed.), Differential Topology, Foliations and Gelfand-Fuks Cohomology, Lecture Notes in Mathematics, vol 652. Springer 1978 (doi:10.1007/BFb0063500)

• Phillip Griffiths, John Morgan, Section 14.5 of Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (1981, 2013) (doi:10.1007/978-1-4614-8468-4)

• Dev Sinha, Ben Walter, Lie coalgebras and rational homotopy theory II: Hopf invariants, Trans. Amer. Math. Soc. 365 (2013), 861-883 (arXiv:0809.5084, doi:10.1090/S0002-9947-2012-05654-6)

• Jean Dieudonné, Section IV.4.B of: A History of Algebraic and Differential Topology, 1900 - 1960, Modern Birkhäuser Classics 2009 (ISBN:978-0-8176-4907-4)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization, Comm. Math. Phys. 2020 (arXiv:1906.07417)