Relative Differential Cohomology

Abstract

We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomology to each other. We discuss long exact sequences for both notions, thereby clarifying their relation to absolute differential cohomology. We construct the external and internal product of relative and absolute characters and show that relative differential cohomology is a right module over the absolute differential cohomology ring. Finally we construct fiber integration and transgression for relative differential characters. 53 Appendix A. Künneth splittings 54 Alexander-Whitney and Eilenberg-Zilber maps. 54 The mapping cone Künneth splitting. 54 References 55 Bibliography 55 Index 572010 Mathematics Subject Classification. 53C08, 55N20.

Relative differential cohomology

“…We have shown in [14] that there are actually four possible inequivalent ways to define the relative Deligne cohomology groups, two of which being the most meaningful and corresponding to the two kinds of differential characters defined in [3,5]. We generalize this picture to any cohomology theory and we show in each case the corresponding long exact sequence.…”

“…It follows that the differential [(F, H , ω)]. 3 • Exactness inĥ n (X ). By construction the restriction to A of a class inĥ n I I (X, A) is topologically trivial.…”

“…In [15,16] a relative version ofĥ • has been defined, considering classes on a pair of manifolds (X, A) whose restriction to A is trivial: nevertheless, when considering the differential refinement of ordinary cohomology, this version is not the only interesting one. It is more natural to consider the case in which the restriction to A is only topologically trivial, so that the curvature is exact but not necessarily vanishing, as in the first definition of relative Cheeger-Simons character given in [5] (see also [2,3]). For example, the B-field in string theory can be described as a Deligne cohomology class of degree 2 in the space-time, which is topologically trivial on a Communicated by Thomas Schick.…”

The Universal -Invariant for Manifolds With Boundary

“…Ad-hoc constructions of relative differential cohomology theories have been considered e.g. in [Fer14] or [Bec13]. But if one represents differential cohomology in terms of sheaves of spectra on the site of smooth manifolds with open covering topology, then the definition of the relative groups becomes completely natural.…”