The Mayer-Vietoris Property in Differential Cohomology

TL;DR

This paper demonstrates that the Mayer-Vietoris property holds for a broad class of differential cohomology functors on compact manifolds, extending previous results specific to complex K-theory.

Contribution

It generalizes the Mayer-Vietoris property to any differential cohomology functor associated with a Z-graded cohomology functor with finitely generated groups, beyond complex K-theory.

Findings

Mayer-Vietoris property holds for all such differential cohomology functors on compact manifolds.

The proof relies on specific commutative diagrams linking different cohomology sequences.

The result broadens the applicability of Mayer-Vietoris in differential cohomology theories.

Abstract

In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differential cohomology functor J^ associated to any Z-graded cohomology functor J(, Z) which, in each degree, assigns to a point a finitely generated group. The approach is to show that the result follows from Diagram 1, the commutative diagram we take as a definition of differential cohomology, and Diagram 2, which combines the three Mayer-Vietoris sequences for J*(, Z), J*(, R) and J*(, R/Z).