Cohomological Invariants of a Variation of Flat Connections

Abstract

Abstract. In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to an r-simplex whose points parametrize flat connections on a smooth manifold X. These invariants lie in degrees (2p − r − 1)-cohomology with C/Z-coefficients, for p > r ≥ 1. This corresponds to a homomorphism on the higher homology groups of the moduli space of flat connections, and taking values in C/Zcohomology of the underlying smooth manifold X.

Invariants of families of flat connections using fiber integration of differential characters

“…Using again a theorem of Bär and Becker [5], we compute a representative differential form. We find that for the p > r + 1 case, this form matches with the one constructed in [4]. Our initial hope was to obtain new cohomology invariants in the p < r case.…”

“…using fiber integration of differential characters. In the next section we show that p > r + 1 case, they agree with the maps constructed in [4]. Our original motivation for doing this construction was to obtain new invariants for the p < r case, however we show in the next section that in this case, the invariants turn out to be trivial.…”

“…She then constructs maps ρ p,r : H r (D(E)) → H 2p−r−1 (B, R/Z) for p > r + 1, r ≥ 1. This article is a humble and modest extension of Jaya Iyer's work (henceforth the article [4] is often referred to as 'Jaya Iyer's paper') using the technique of fiber integration of differential characters developed by Bär and Becker in [5]. Given an element u compatible with P as above, we derive maps ψ P,u,r : H r (D(E)) → H 2p−r−1 (B, R/Z) for p = r, r + 1.…”

“…In this article, we consider the problem of attaching invariants to a family of flat connections. More precisely, we formulate the problem as done in [4] : Let D(E) be the simplicial abelian group, whose r-simplices are freely generated by (r + 1)−tuples (D 0 , D 1 , · · · , D r ) of relatively flat connections on the bundle E → B. (The connections D 0 , D 1 , · · · , D r are called relatively flat if the linear combination t 0 D 0 + t 1 D 1 + · · · + t r D r is a flat connection for each choice of t 0 , · · · , t r such that…”

“…Thus the authors determined R/Z cohomology classes which we denote by cs (P,u) (E, θ) ∈ H 2p−1 (B, R/Z). Generalizing this work to families of connections, Jaya Iyer [4] showed how to associate an element of H 2p−r−1 (B, R/Z) to an element of r-th simplicial homology of the simplicial abelian group of relatively flat connections on the bundle. More precisely, she defines the simplicial abelian group D(E) whose set of r−simplices is the free abelian group generated by (r + 1)tuples (D 0 , · · · , D r ) of relatively flat connections.…”