Amiya Mukherjee

An axiomatic approach to equivariant cohomology theories

This paper presents a translation of a theorem of Cartan into an equivariant setting. This work is largely based on the study of the homotopical algebra in the sense of Quillen of the category of simplicial objects over the category of rationalOg-vector spaces. The application is a solution to the equivariant commutative cochain problem. This solution is slightly better than the solution obtained earlier by Triantafillou in that the transformation groupG need not be finite.

Higher-order nondegenerate immersions of manifolds

Abstract In this paper we show how the Smale-Hirsch theory of immersions can be adapted to get a simple proof of an integrability theorem of Gromov and Eliashberg concerning higher-order nondegenerate immersions.

Basic Concepts of Manifolds

There are two ways one can look at a differentiable manifold. Firstly, it is a topological space with a structure which helps us to define differentiable functions on it, just as a topological structure on a set is designed to define continuous functions on that set. Secondly, it is a topological space which can be obtained by gluing together open subsets of some Euclidean space in a nice way; think, for example, of the surface of a ball or a torus covered with small paper disks pasted together on overlaps without making any crease or fold.

Linear Structures on Manifolds

The title of the chapter may seem to be all embracing, as almost any thing of the manifold theory may come under this heading. In fact, there is hardly any structure on manifolds that is not influenced by linear algebra. However, we have selected among others two structures on manifolds, namely, the symplectic and contact structures.

Vector Bundles on Manifolds

The theory of vector bundles provides a very elegant and concise language to describe many phenomena in manifolds. In its simplest situation, a vector bundle over a manifold M is a manifold E which is the disjoint union of a family of vector spaces \(\left\{E_x\right\}\), indexed by \(x\;\in\;M\).

The Ricci Flow Equation and Poincaré Conjecture

The Poincare conjecture was formulated by the French mathematician Henri Poincare more than hundred years ago. The conjecture states, when reformulated in modern language, that any simply connected closed 3-manifold is diffeomorphic to the standard 3-sphere \(S^3\). This was the most famous open problem, and its solution turned out to be extraordinarily difficult. It had eluded all attempts at solution for more than hundred years. During 2002 and 2003, Grigoriy Perelman posted a proof of the conjecture on the Internet in three instalments, completing a program initiated in the 1980s by Richard Hamilton to solve a more general conjecture, called the geometrization conjecture of William Thurston. The key tool of Hamilton’s program is the Ricci flow, a differential equation on the space of Riemannian metrics of a 3-manifold. The equation is designed after the mathematical model for heat flow. As heat gradually flows from hotter to cooler parts of a metallic body until a uniform temperature is achieved throughout the body, it was expected that in Ricci flow, regions of higher curvature will tend to diffuse into regions of lower curvature to produce an equilibrium geometry for the 3-manifold for which Ricci curvature is uniform over the entire manifold. Thus in principle, a 3-manifold when subject to Ricci flow will produce a kind of normal form which will ultimately solve the geometrization conjecture. Although Hamilton established a number of beautiful geometric results using the Ricci flow equation, the progress in applying this program to the conjecture eventually came to a standstill mainly because of the formation of singularities, which defied solution of the problem. In his proof, Perelman constructed a program for getting around to these obstacles. He modified the Ricci flow used by Hamilton with “Ricci flow with surgery”. This expunges the singular regions as they develop in a controlled way and eventually solves the geometrization conjecture.

Approximation Theorems and Whitney’s Embedding

Perhaps the most important property of a manifold which opens up various developments of manifold theory is that a manifold can be embedded in a Euclidean space as a closed subspace. This is called Whitney’s embedding theorem. Thus any manifold may be considered as a submanifold of a Euclidean space.

Spaces of Smooth Maps

Many problems of differential topology can be formulated as problems about spaces of smooth maps between manifolds, and their associated jet spaces. For example, the problems of transversality which we considered earlier can be rephrased in terms of jets. This chapter is devoted to topics and results related to spaces of maps and jet spaces. The main aim is to prove Thom’s transversality theorem, and find some of its applications. One of the results gives a better perspective on Whitney’s embedding theorem which we have proved already in Chapter 2.

Theory of Handle Presentations

All manifolds that will appear in this chapter will be compact unless stated otherwise. Our first result is the handle presentation theorem which says any cobordism M admits a Morse function f, and a decomposition, which is obtained by successively attaching handles corresponding to the critical points of f.

Cohomological Formulation of the Index Theorem

We have proved the splitting principle for complex bundles in Theorem 5.1.3. The splitting principle for real vector bundles states that