Raoul Bott

Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections

This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self‐linking integrals of Guadaguini, Martellini, and Mintchev [‘‘Perturbative aspects of Chern–Simons field theory,’’ Phys. Lett. B 227, 111 (1989)] and Bar‐Natan [‘‘Perturbative aspects of the Chern–Simons topological quantum field theory,’’ Ph.D. thesis, Princeton University, 1991; also ‘‘On the Vassiliev Knot Invariants’’ (to appear in Topology)] are seen to represent the first nontrivial element in H0(F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

Morse theory indomitable

At present a great deal is known about the value distribution of systems of meromorphic functions on an open Riemann surface. One has the beautiful results of Picard, E. Borel, Nevanlinna, Ahlfors, H. and J. Weyl and many others to point to. (See [1], [2].) The aim of this paper is to make the initial step towards an n-dimensional analogue of this theory. A natural general setting for the value distribution theory is the following one. We consider a complex n-manifold X and a holomorphic vector bundle E over X whose fiber dimension equals the dimension of X and wish to study the zero-sets of holomorphic sections of E. When X is compact (and without boundary) then it is well-known that if the zeroes of any continuous section are counted properly then the algebraic sum of these zero-points is independent of the section and is given by the integral of the nth Chern(2) class of E over X: Thus we have zeroes of s = | cn(E), (1.1) Number of J x

Integral invariants of 3-manifolds. II

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On the de Rham theory of certain classifying spaces

This note is a sequel to our earlier paper of the same title [4] and describes invariants of rational homology 3-spheres associated to acyclic orthogonal local systems. Our work is in the spirit of the Axelrod–Singer papers [1], generalizes some of their results, and furnishes a new setting for the purely topological implications of their work.

On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups

In this thesis, H. Shulman proved the vanishing phenomena for characteristic classes of foliations without the usual geometric constructions of connections, curvature, etc. In this note, we present a setting for this, which was noted independently by the senior authors about two years ago. Since that time, this subject has grown considerably, so that our account here is to a large extent anachronistic. In particular, the work of Kamber and Tondeur, who combine simplicial and curvature techniques but avoid classifying spaces, as well as the work of Vey, Bott, Haefliger and others has progressed far beyond the results outlined here. Nevertheless, our ideas are on the one hand very simple, but on the other, involve technicalities that have been resolved only recently, that a short account at this time still seems worthwhile. The basic concept, which has been in the air for quite some time, and notably in the work of Deligne on mixed Hodge structures, is that the de Rham theory can be profitably employed as a tool for studying certain nonmanifolds, namely, those that are obtained as the geometric realization of a simplicial manifold.

On the periodicity theorem for complex vector bundles

This note describes a formula found in collaboration with G. Hochschild during the spring of 1972 at Berkeley, California. Therefore I would like first to thank him for all he taught me during those months, and, second, to chide him for steadfastly refusing to share the blame with me as a joint author. Since that time independent work of Kamber Tondeur has also appeared [6], which covers much of this material but from a rather different point of view. (Thus Theorem 2.10 of their Manuscripta paper is essentially equivalent to our Theorem 1.) The aim of our discussion was really to explain the Chern-Weil homomorphism to topologists, and I hope that in this respect we have succeeded. The present point of view also fits very well into the general framework of foliations and the continuous cohomology encountered there, as will be seen in a joint paper now in progress with A. Haefliger. However, in this note I will stick to the finite-dimensional case, and essentially start from scratch. To explain our formula recall first of all that the Chern-Weil homomorphism links the cohomology of the classifying space BG of a Lie-group, G, to the invariant forms on the Lie algebra g of G. More precisely, if g* denotes the dual of g as a G-module under the adjoint action, and Sg* the symmetric algebra on g* in its induced G-module structure, then the Chern-Weil construction defines a homomorphism from the invariants Invc(Sg*) of Sg* to H*(BG) = H*(BG; R) rp: Inv&Sg*) + H*(BG). (1.1) The construction of this q was inspired by differential geometric considerations and generalizes earlier constructions for the Pontryagin

A Lefschetz fixed point formula for elliptic differential operators

where K(X) is the Grothendieck group (2) of complex vector bundles over X. The general theory of these K-groups, as developed in [1], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof. Our proof will be strictly elementary. To emphasize this fact we have made the paper entirely self-contained, assuming only basic facts from algebra and topology. In particular we do not assume any knowledge of vector bundles or K-theory. We hope that, by doing this, we have made the paper intelligible to analysts who may be unacquainted with the theory of vector bundles but may be interested in the applications of K-theory to the index problem for elliptic operators [2]. We should point out in fact that our new proof of the periodicity theorem arose out of an attempt to understand the topological significance of elliptic boundary conditions. This aspect of the matter will be taken up in a subsequent paper.(3) In fact for the application to boundary problems we need not only the periodicity theorem but also some more precise results that occur in the course of our present proof.

On the Zeroes of Meromorphic Vector-Fields

Introduction. The classical Lefschetz fixed point formula expresses, under suitable circumstances, the number of fixed points of a continuous map ƒ : X-+X in terms of the transformation induced by ƒ on the cohomology of X. If X is not just a topological space but has some further structure, and if this structure is preserved by ƒ, one would expect to be able to refine the Lefschetz formula and to say more about the nature of the fixed points. The purpose of this note is to present such a refinement (Theorem 1) when X is a compact differentiable manifold endowed with an elliptic differential operator (or more generally an elliptic complex). Taking essentially the classical operators of complex and Riemannian geometry we obtain a number of important special cases (Theorems 2,3) . The first of these was conjectured to us by Shimura and was proved by Eichler for dimension one.