Johan L. Dupont

Curvature and characteristic classes

Differential forms and cohomology.- Multiplicativity. The simplicial de rham complex.- Connections in principal bundles.- The chern-weil homomorphism.- Topological bundles and classifying spaces.- Simplicial manifolds. The chern-weil homomorphism for BG.- Characteristic classes for some classical groups.- The chern-weil homomorphism for compact groups.- Applications to flat bundles.- Errata.

SIMPLICIAL DE RHAM COHOMOLOGY AND CHARACTERISTIC CLASSES OF FLAT BUNDLES

IT IS well-known that the classifying space BG of a Lie group G is the geometric realization of a simplicial manifold, i.e. a semi-simplicial set whose p-dimensional simplices constitute a C” manifold and whose boundary and degeneracy operators are C” maps (see e.g. Segal[231). In the study of characteristic classes in real cohomology it is therefore natural to look for a De Rham complex for a simplicial manifold X = {X,}. An obvious candidate is the total complex a*(X) of the double complex (aq(X,), 6, &) of C” q-forms on X,, where dx is the usual exterior differential and where 6 is the co-boundary of simplicial cochains. This is studied in a recent paper by Bott-Shulman-Stasheff [4], where one can find a proof of the fact that the homology H(&*(X)) is naturally isomorphic to the singular cohomology with real coefficients of the realization ]]X]]. However there is an even more natural De Rham complex A *(X) associated to a simplicial manifold where a “form” is roughly speaking a C” form on llX]l (see 02 for a precise definition). For X a discrete simplicial set the construction of A *(X) goes back to Whitney [26] and has recently been used by Sullivan[25] in his study of the rational homotopy type of a manifold. The advantage of A *(X) is (apart from the suggestive nature of the definition) that the multiplication is graded commutative as in the case of an ordinary manifold and so the usual Chern-Weil theory carries over word by word to the universal case X = NG, the nerve of a Lie group G. To a great extent this just leads to a reformulation of previous constructions by Bott and Shulman (see [2] and [3]) and by Kamber-Tondeur[l3]. However the present point of view gives rise to an interesting formula for the characteristic classes of flat bundles which we shall now describe. Let G be a connected semi-simple real Lie group with finite center and choose a maximal compact subgroup K 2 G. Let fi and I be the corresponding Lie algebras and let H*(g, f) be the relative Lie algebra cohomology. For a homomorphism f: I + G where I is a discrete group there is a well-known characteristic homomorphism jr: H*(g, f)+ H*(Br) whose definition via De Rham cohomology goes back to Matsushima[ 171 (see Kamber-Tondeur[ 131,

Scissors congruences, II

8 or

Scissors congruences, group homology and characteristic classes

4 below) and which has recently been studied by Bore1 [l] for arithmetic subgroups I. Now H*(Br) is canonically isomorphic to H*(T) the Eilenberg-MacLane group cohomology of I with real coefficients (see e.g. MacLane [ 161, chapter 4, P5) and we want to express jr in terms of explicit cochains. Let g = p @ f be a Cartan decomposition (see e.g. Helgason[ 101, chapter 3, 57). Then a class in H’(fl, I) is represented by an alternating q-form cp on g/f = p. By left translation this gives a closed C” q-form 6 on G/K (the differential of the complex A*(p) is actually 0 since [p, p] c f). Endowed with a left invariant Riemannian metric G/K is a non-compact globally symmzric space and I acts via f as a group of isometries on G/K (so we shall write -+x instead of f(y)x for x E G/K and y E I). Now let (r,, . . . , yq) E (r)’ and let o = {K} E G/K be the base point. We define the geodesic simplex A(y,, . . . , yq) g G/K inductively as follows. A(-y,) is the geodesic arc from o to ylo and generally A(r,, . . . , yq) is the geodesic cone on yl . A(?*, . . . , yq) with top point o. The ordering of the vertices o, y,o, y,-yZo, . . . , ylyz . . . yqo, determines a natural orientation of A(?,, . , . , yq). In this notation we shall prove (54):

Homology of classical Lie groups made discrete. II. H2, H3, and relations with scissors congruences☆

In general, Y(X) is the scissors congruence group of polytopes in the space X. Unless stated explicitly, the group of motions of X is understood to be the group of all isometries of X. _@ is the extended hyperbolic n-space; it is obtained by adding to the hyperbolic n-space x”’ all the ideal points lying on 65~~. The geometry of a.F is that of conformal geometry on a sphere of dimension n 1. The group 4~~) captures the scissors congruence problem in a precise manner. On the other hand, the stable scissors congruence group p(_

A dilogarithmic formula for the Cheeger-Chern-Simons class

‘“) is more maneuverable, see Sah [19].

Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences

Introduction and history scissors congruence group and homology homology of flag complexes translational scissors congruences Euclidean scissors congruences Sydlers theorem and non-commutative differential forms spherical scissors congruences hyperbolic scissors congruences homology of Lie groups made discrete invariants simplices in spherical and hyperbolic 3-space rigidity of Cheeger-Chern-Simons invariants projective configurations and homology of the projective linear group homology of indecomposable configurations the case of PGI(3,F).

Homology of O(n) and O1(1,n) made discrete: An application of edgewise subdivision

The present work extends a number of earlier results; see [S, 7, 15-181. For example, the following fundamental exact sequence (essentially due to Bloch and Wigner in somewhat different form, but not published by them) can be found in DuPont and Sah [7]: o~o/z-tH3(SL(2,@))~~(@)~n:(a=~)~~,(a=)~o. (2.12) The group Y(C) is defined in (1.4))( 1.8) and is isomorphic to the group PC considered in DuPont and Sah [7]. Also, the study of the scissors congruence problem had led to two exact sequences in DuPont [S, Theorems 1.3 and 1.43 that roughly correspond to the f -eigenspaces of (2.12) under the action of complex conjugation: O~A~H,(SU(2))~~S3/L~[WO([W/Z)-*H,(SU(2))-tO; (0.1)

Gerbes, Simplicial Forms and Invariants for Families of Foliated Bundles

We present a simplification of Neumann’s formula for the universal Cheeger‐ Chern‐Simons class of the second Chern polynomial. Our approach is completely algebraic, and the final formula can be applied directly on a homology class in the bar complex. 57M27; 57T30

Three Questions about Simplices in Spherical and Hyperbolic 3-Space

The present work continues our investigations on the Scissors Congruence Problems. These investigations originated with the Third Problem of Hilbert that dealt with the scissors congruence problem in Euclidean 3-space. As indicated in If], [5], [16], the non-Euclidean versions are just as interesting. They have an intimate connection with the Eilenberg-MacLane homology of certain classical Lie groups (namely, the isometry groups of the appropriate classical geometries) and come into contact with algebraic Ktheory, Cheeger-Chern-Simons characteristic classes, as well as other topics. In all three series of classical geometries, the spherical version enters because the basic Dehn invarients require an understanding of the spherical scissors congruence problem. In a number of recent works, we have concentrated our efforts on the non-Euclidean cases, see [6], [18] for results and summaries in these directions. In spite of our efforts, the most complete results remain to be the theorems of Dehn-Sydler-Jessen showing that volume and Dehn invariants form a complete system of invariants for the scissors congruence problem in Euclidean spaces of dimensions 3 and 4. The original work of Sydler [19] was an incredible tour de force geometric arument in Euclidean 3-space. It was rapidly simplified by Jessen in [9] and extended to Euclidean 4-space. The simplification by Jessen employed techniques from homological algebra. Nevertheless, two of the geometric arguments of Sydler were retained in Jessens work. The present work continues in the direction of the general theme that the scissors congruence problems should be formulated and solved in terms of the Eilenberg-MacLane homology of classical groups (with appropriate coefficients). The principal goal in the present