An Introduction to Intersection Homology Theory: 2nd Edition (Hardback) book cover

An Introduction to Intersection Homology Theory

2nd Edition

By Frances Kirwan, Jonathan Woolf

Chapman and Hall/CRC

248 pages | 10 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9781584881841
pub: 2006-06-07
SAVE ~$14.24
Currently out of stock
$94.95
$80.71
x


FREE Standard Shipping!

Description

Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory.

Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans.

Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.

Table of Contents

INTRODUCTION

Poincaré duality

Morse theory for siningular spac es

de Rham cohomology and L2-c -cohomol ology

The cohomology of pr projective vari ties

REVIEW OF HOMOLOGY AND COHOMOLOGY

Simplicial homology

Singular homology

Homology with close closed support

Conclusion

Further reading

REVIEW OF SHEAF COHOMOLOGY AND DERIVED CATEGORIES

Sheaves

Cech cohomology of sheaves

Hypercohomology

Functors and exactness

Resolution of sheaves of complexes

Cohomology and hypercohomology via derived functors

Derived categories.

Right derived functors

Further reading.

THE DEFINITION OF INTERSECTION HOMOLOGY

Stratified spaces and pseudomanifolds

Simplicial intersection homology

Singular intersection homology

Simple examples of intersection homology

Normalisati ons

Relative groups and the Mayer-Vietoris sequence.

The intersection homology of a cone

Functoriality of intersection homology

Homology with local coefficients

Quasi-projec tive complex varieties

Further reading

WITT SPACES AND DUALITY

Generalised Poincaré duality.

Witt spaces

Signatures of Witt spaces

The Witt-bordism groups

Further reading

L2- COHOMOLOGY AND INTERSECTION ON COHOMOLOGY

L2-cohomology and Hodge theory

The L2-cohomology of a punctured cone

Varieties with isolated conical singularities

Locally symmetric varieties

Further reading.

SHEAF-THEORETIC INTERSECTION HOMOLOGY

Sheaves of singular chains

Constructibility and an axiomatic characterisation

The topological invariance of intersection homology

Duality in the derived category

Further reading

PERVERSE SHEAVES

Perverse sheaves

Perverse sheaves on varieties

Nearby and vanishing cycles

The decomposition theorem

Further reading

THE INTERSECTION COHOMOLOGY OF FANS

Affine toric varieties

Toric varieties from fans

Maps and torus actions

Projective toric varieties and convex polytopes

Stratifications of toric varieties

Subdivisions and desingularisations

Equivariant intersection cohomology

The intersection cohomology of fans

Stanley's conjectures

Further reading

CHARACTERISTIC p AND THE WEIL CONJECTURES

Statement of the Weil conjectures

Basic properties of ,-adic cohomology

Étale topology and cohomology

The Weil conjectures for singular varieties

Further reading

D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE

The Riemann-Hilbert problem

Differential systems over Cn

Dx-modules and intersection homology

The characteristic variety of a Dx-module

Holonomic differential systems

Examples of characteristic varieties

Left and right Dx-modules

Restriction of Dx-modules

Regular singularities

The Riemann-Hilbert correspondence

Further reading

THE KAZHDAN-LUSZTIG CONJECTURE

Verma modules

D-modules over flag manifolds

Characteristic p

Hecke algebras and the Kazhdan-Lusztig polynomials

Further reading

Bibliography

Index

Subject Categories

BISAC Subject Codes/Headings:
MAT012000
MATHEMATICS / Geometry / General
MAT022000
MATHEMATICS / Number Theory