Relative differential characters (Q856792)

Given a principal \(G\)-bundle with connection on a smooth manifold \(M\), an invariant polynomial gives rise to a Chern-Weil form in the base. Lifting this to the total space gives an exact form and the well known form of Chern and Simons has the property that its differential is this exact form. If one happens to have a global section, one can pull this back to the base space, but in general Chern-Simons forms live in the total space. Cheeger and Simons introduced the notion of a differential character as a way of understanding Chern-Simons forms in terms of the base space. These differential characters can be assembled into a group \(\hat H^k(M)\) which contains both homotopical and geometrical information. One may also think of differential characters as encoding how integral cycles and differential forms with integral periods interact when viewed in real cohomology. When \(k=2\), the Cheeger-Simons group classifies \(U(1)\)-bundles with connection and when \(k=3\), it classifies \(U(1)\)-gerbes with connection. The latter makes Cheeger-Simons groups relevant to ``stringy'' topology since a gerbe with connection can be interpreted as a line bundle over a free loop space with parallel transport over surfaces. In this paper, the authors studied the notion of relative differential character associated to a pair of manifolds \(A\) and \(M\) together with a smooth map \(\rho:A\rightarrow M\). The authors formulate two competing definitions for the relative group. The first is a natural extention of Cheeger's and Simons' work giving what is called the relative Cheeger-Simons group. The second follows the more recent work of Hopkins and Singer defining the Cheeger-Simons group as the cohomology of a certain cochain complex, giving what it called the relative Hopkins-Singer group. The purpose of this paper is to clarify the relationship between these two groups. Unlike the non-relative case, the two models are not equivalent. The authors showed that the relative Cheeger-Simons model sits in three short exact sequences analogous to the ones occupied by ordinary differential characters. They also showed that the relative Hopkins-Singer model sits in a long exact sequence analogous to the one occupied by relative de Rham cohomology. Finally, the authors related the two models. In particular, they proved that the relative Hopkins-Singer model is equivalent to a quotient of a subgroup of the relative Cheeger-Simons model.