group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For odd the integral Steenrod square is the composition of the mod-2 Steenrod square with the Bockstein homomorphism associated with the sequence :
The odd-degree integral Steenrod squares from def. are indeed integral lifts of the mod-2 Steenrod squares in that
This follows from the relation of the Bockstein homomorphism to the first Steenrod square
(this example) together with the first Adem relation
(this example):
(integral Steenrod square in terms of Bockstein homomorphism for exponential sequence)
The integral Steenrod squares (def. ) may equivalently be written in terms of the Bockstein homomorphism of the exponential sequence as
Since , by this example.
(integral Steenrod square refined to ordinary differential cohomology)
Let be a cocycle in ordinary differential cohomology of degree .
By inserting the bottom triangle of the ordinary differential cohomology hexagon (this diagram) into the factorization in (1) we obtain a canonical refinement of the integral Steenrod square to a cocycle in ordinary differential cohomology, which happens to be flat
If one moreover asks that the integral Steenrod square vanishes
(as in Diaconescu-Moore-Witten 00, around (6.9) for ) then the curvature exact sequence and characteristic class exact sequence in ordinary differential cohomology (this prop.) imply that the class of is identified with a class in de Rham cohomology in degree :
The third integral Steenrod square plays a central role in the discussion of the supergravity C-field in
Last revised on November 19, 2020 at 22:39:15. See the history of this page for a list of all contributions to it.