The Mayer-Vietoris Property in Differential Cohomology

James Simons, Dennis Sullivan

Published 2010-10-25, updated 2010-11-01Version 2

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).