nLab flat K-theory




In the context of topological K-theory (at least), flat K-theory (first considered in Karoubi 1987, §7.21, 1990, Ex. 3) is the Whitehead-generalized cohomology theory which is (represented by) the homotopy fiber of the Chern character; equivalently the restriction (namely the homotopy pullback) of differential K-theory (any version of it) to vanishing generalized-curvature.

Since the coefficients of flat KU form the circle group U(1)/U(1) \simeq \mathbb{R}/\mathbb{Z}, flat K-theory is also known as /\mathbb{R}/\mathbb{Z}-K-theory or as K-theory with /\mathbb{R}/\mathbb{Z}-coefficients (Lott 1994), or similar.

In the differential cohomology hexagon for topological K-theory, flat K-theory appears in outer long exact sequence (Simons & Sullivan 2008, p. 2):

(graphics from SS22)


Original accounts:

Discussion as a vertex in the differential cohomology hexagon:

Created on June 13, 2022 at 17:26:37. See the history of this page for a list of all contributions to it.