An introduction to Characteristic Classes

✍ Scribed by Jean-Paul Brasselet

• Publisher: IMPA
• Year: 2021
• Tongue: English
• Leaves: 136
• Edition: 1
• Category: Library

✦ Table of Contents

Preface
Introduction
Manifolds and pseudomanifolds
Orientation
Orientation of pseudomanifolds
Orientation of manifolds
Oriented double covering
Poincaré isomorphism (manifolds)
Boundary
The Gauß map(s)
Generalisation of the Gauß map
Fibre bundles.
Fibre bundles
Vector bundles
Examples of fibre bundles – real case
Examples of fibre bundles – complex case
The ``Euler–Poincaré'' characteristic
The Greek period
Maurolico – Descartes – Euler
Maurolico (1494–1575)
Descartes (1596–1650)
Euler (1707–1783)
Descartes' Theorem is equivalent to Euler Formula
Proofs of Euler Formula
The generalization: Euler–Poincaré characteristic
Poincaré–Hopf Theorem
The index of a vector field.
Relation with the Gauß map
Proof of Poincaré–Hopf Theorem
The smooth case without boundary
The smooth case with boundary
Characteristic classes : smooth case
General obstruction theory
Index of an r-frame
General obstruction theory
Applications
Stiefel–Whitney classes
Chern classes
Singular varieties
Stratifications
Angles
Whitney stratifications
Fundamental properties of Whitney stratifications
Poincaré homomorphism
Alexander isomorphism
Cellular tubes
Poincaré–Hopf Theorem (singular varieties)
Introduction
Why the radial vector fields ?
Why the dual cells decomposition ?
Radial vector fields
Radial vector fields – Local construction
Radial vector fields – Global construction
Poincaré–Hopf Theorem for singular varieties.
Schwartz classes
Radial extension of frames
Local radial extension of r-frames
Global radial extension of r-frames
Obstruction cocycles and classes
MacPherson classes
Nash transformation
local Euler obstruction
Constructible sets and functions
Mather classes
MacPherson classes
Developments and perspectives
Remarks and complements.
About the fundamental Chern article.
The polar varieties and Mather classes
More developments of Chern classes for singular varieties
Bivariant classes
Other generalizations of classes in the singular case.
Hirzebruch formalism
The Euler local obstruction
Some applications in other mathematical domains and in theoretical physics.
Bibliography
Index of Authors
Index