2nd Edition
An Introduction to Intersection Homology Theory
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.
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
