Universal Characteristic Classes Research Articles

Cheeger–Chern–Simons Theory and Differential String Classes

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class $${\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}$$ , we recover isomorphism classes of geometric string structures on Spin n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class $${\frac{1}{2} p_1}$$ together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.

$G$-gerbes, principal $2$-group bundles and characteristic classes

Let $G$ be a Lie group and $G\to\Aut(G)$ be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group $[G\to\Aut(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e.\ between principal $[G\to\Aut(G)]$-bundles over differentiable stacks and $G$-gerbes over differentiable stacks). This approach also allows us to identify $G$-bound gerbes and $[Z(G)\to 1]$-group bundles over differentiable stacks, where $Z(G)$ is the center of $G$. We also introduce universal characteristic classes for 2-group bundles. For groupoid central $G$-extensions, we introduce Dixmier--Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier--Douady classes are integral.

Chern–Weil map for principal bundles over groupoids

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman map $$S(\mathfrak{g}^*)^G \to H^*(BG)$$ when the manifold is a point.

The index of an equivariant vector field and addition theorems for Pontryagin classes

In this paper we construct a theory of indices of Morse-Bott vector fields on a manifold and use it to solve a famous localization problem for the transfer map. As a consequence, we obtain addition theorems for universal Pontryagin classes in cobordisms. This enables us to complete the construction of the theory of universal characteristic classes, which was begun more than twenty years ago.

Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces BK where K is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form. This work solves a problem posed by Weinstein, who gave a corresponding construction (following Karshon and Goldman) of the symplectic forms on these moduli spaces. We also give a corresponding construction of equivariant differential forms on the extended moduli space X, which is a finite dimensional symplectic space equipped with a Hamiltonian action of K for which the symplectic reduced space is the moduli space of representations of the 2-manifold fundamental group in K.

On the linear independence of certain cohomology classes in the classifying space for subfoliations

The purpose of this paper is to establish the linear independence of certain cohomology classes in the Haefliger classifying space BF(q, , q2) for subfoliations of codimension (q1, q2). The classes considered are of secondary type, not belonging to the subalgebra of H(B1T(q, q2), R) generated by the union of the universal characteristic classes for foliations of codimension q1 and q2 respectively, and are elements of the kernel of the canonical homomorphism H(BF(q ,q2), R) H(BFqj x B1d, R) with d = q2q > 0.

Cohomology of split group extensions, II

They introduced a spectral sequence relating the cohomology of G (with a suitable filtration) to the cohomologies of G/K and K. In later developments, Charlap and Vasquez [5, 61 showed that, under suitable restrictions, d, of the Hochschild-Serre spectral sequence can be described as the cup product with certain “characteristic” cohomology classes. In the special case where K is a free abelian group L of finite rank N and when G splits over K by r = G/K, these characteristic classes v 2t lie in H2(r, fP1(L, H,(L, Z))). Through straightforward, though formidable, procedures, Charlap and Vasquez found a number of interesting results concerning these characteristic classes. However, these characteristic classes still appeared mysterious and difficult to handle. The purpose of the present paper is to present a procedure to determine the most important one of these characteristic classes. To be precise, Charlap and Vasquez showed that vZt E H2(F, fPl(L, H,(L, Z))) have orders dividing 2; vsl = 0 and vpt = 0 holds for all t if and only if v22 = 0. They also showed that vZt are functorial in r so that vZt are universal when .F = GL(N, Z). We show that H2(GL(N, Z), HI(L, H,(L, Z))) is a group of order 2 for N > 2, and when N > 2, its generator is the universal characteristic class. Along the way, we calculate a number of cohomology groups similar to the one described. Our results and procedures can be applied to study the torsion of the total space of certain fibered varieties over symmetric space with abelian varieties for fibers (cf. Kuga [14]). Th ese applications will appear in a subsequent paper. Some of the cohomology group calculations are also useful in the study of finite simple groups. Indeed, one of the phenomena that persists throughout