Singular Intersection Homology

Contents
Preface
Notations and Conventions
	Spaces
	Algebra
	Algebraic topology
	Intersection homology and cohomology
	Miscellaneous conventions
1. Introduction
	1.1 What Is Intersection Homology?
	1.2 Simplicial vs. PL vs. Singular
	1.3 A Note about Sheaves and Their Scarcity
	1.4 GM vs. Non-GM Intersection Homology
	1.5 Outline
2. Stratified Spaces
	2.1 First Examples of Stratified Spaces
	2.2 Filtered and Stratified Spaces
		2.2.1 Filtered Spaces
		2.2.2 Stratified Spaces
			(a) Manifold Stratified Spaces
		2.2.3 Depth
	2.3 Locally Cone-like Spaces and CS Sets
	2.4 Pseudomanifolds
	2.5 PL Spaces and PL Pseudomanifolds
		2.5.1 PL Spaces
		2.5.2 Piecewise Linear and Simplicial Pseudomanifolds
			(a) Classical Simplicial Pseudomanifolds
	2.6 Normal Pseudomanifolds
	2.7 Pseudomanifolds with Boundaries
	2.8 Other Species of Stratified Spaces
		2.8.1 Whitney Stratified Spaces
		2.8.2 Thom–Mather Spaces
		2.8.3 Homotopically Stratified Spaces
	2.9 Maps of Stratified Spaces
	2.10 Advanced Topic: Intrinsic Filtrations
		2.10.1 Intrinsic PL Filtrations
			(a) Intrinsic Filtrations of PL Pseudomanifolds with Boundary
	2.11 Advanced Topic: Products and Joins
		2.11.1 Products of Intrinsic Filtrations
3. Intersection Homology
	3.1 Perversities
		3.1.1 GM Perversities
		3.1.2 Dual Perversities
	3.2 Simplicial Intersection Homology
		3.2.1 First Examples
			(a) Allowability with Respect to Regular Strata
			(b) Effects of Subdivision
			(c) Some More Advanced Examples
		3.2.2 Some Remarks on the Definition
			(a) The Motivation for the Definition of Intersection Homology
			(b) Strata vs. Skeleta in the Definition of Intersection Chains
	3.3 PL Intersection Homology
		3.3.1 PL Homology
			PL Chains and PL Maps
		3.3.2 A Useful Alternative Characterization of PL Chains
			(a) Adding Chains
			(b) Compatibility with PL Maps
			(c) Realization
		3.3.3 PL Intersection Homology
		3.3.4 The Relation between Simplicial and PL Intersection Homology
	3.4 Singular Intersection Homology
4. Basic Properties of Singular and PL Intersection Homology
	4.1 Stratified Maps, Homotopies, and Homotopy equivalences
	4.2 The Cone Formula
	4.3 Relative Intersection Homology
		4.3.1 Further Commentary on Subspace Filtrations
		4.3.2 Stratified Maps Revisited
		4.3.3 Reduced Intersection Homology and the Relative Cone Formula
			(a) Reduced Intersection Homology
			(b) The Relative Cone formula
	4.4 Mayer–Vietoris Sequences and Excision
		4.4.1 PL Excision and Mayer–Vietoris
		4.4.2 Singular Subdivision, Excision, and Mayer–Vietoris
			(a) Singular Subdivision
			(b) Excision
			(c) Mayer–Vietoris
			(d) Examples
			(e) Relative Mayer–Vietoris sequences
5. Mayer–Vietoris Arguments and Further Properties of Intersection Homology
	5.1 Mayer–Vietoris Arguments
		5.1.1 First Applications: High Perversities and Normalization
			(a) High Perversities
			(b) Normalization
	5.2 Cross Products and the Künneth Theorem with a Manifold Factor
		5.2.1 The Singular Chain Cross Product
		5.2.2 The PL Cross Product
		5.2.3 Properties of the Cross Product
		5.2.4 Künneth Theorem when One Factor Is a Manifold
	5.3 Intersection Homology with Coefficients and Universal Coefficient Theorems
		5.3.1 Definitions of Intersection Homology with Coefficients
			(a) Comparing the Options
			(b) Basic Properties of Intersection Homology with Coefficients
		5.3.2 Universal Coefficient Theorems
	5.4 Equivalence of PL and Singular Intersection Homology on PL CS Sets
		5.4.1 Barycentric Subdivisions and Maps from PL Chains to Singular Chains
		5.4.2 The Isomorphism of PL and Singular Intersection Homology
	5.5 Topological Invariance
		5.5.1 Which Perversities Work?
		5.5.2 The Statement of the Theorem and Some Corollaries
		5.5.3 Proof of Topological Invariance
	5.6 Finite Generation
6. Non-GM Intersection Homology
	6.1 Motivation for Non-GM Intersection Homology
	6.2 Definitions of Non-GM Intersection Homology
		6.2.1 First Definition of IH∗
		6.2.2 Second Definition of IH∗
		6.2.3 Third Definition of IH∗
		6.2.4 Non-GM Intersection Homology below the Top Perversity
		6.2.5 A New Cone Formula
		6.2.6 Relative Non-GM Intersection Homology and the Relative Cone Formula
	6.3 Properties of I^{¯p}H∗ (X;G)
		6.3.1 Basic Properties
			(a) Maps and Homotopies
			(b) Subdivision, Excision, and Mayer–Vietoris
			(c) Applications of Mayer–Vietoris Arguments
			(d) Cross Products
			(e) Coefficients
			(f) Agreement of Singular and PL Intersection Homology
			(g) Finite Generation
		6.3.2 Dimensional Homogeneity
		6.3.3 Local coefficients
	6.4 A General Künneth Theorem
		6.4.1 A Key Example: the Product of Cones
			(a) Necessity of Our Conditions on Q
			(b) Sufficiency of Our Conditions on Q
			(c) The Cross Product
		6.4.2 The Künneth Theorem
		6.4.3 A Relative Künneth Theorem
		6.4.4 Applications of the Künneth Theorem
		6.4.5 Some Technical Stuff: the Proof of Lemma 6.4.2
			(a) Algebra of the Algebraic Künneth Theorem
			(b) Intersection Homology Products with Cones
	6.5 Advanced Topic: Chain Splitting
7. Intersection Cohomology and Products
	7.1 Intersection Cohomology
	7.2 Cup, Cap, and Cross Products
		7.2.1 Philosophy
		7.2.2 Intersection Homology Cup, Cap, and Cross Products
			(a) Hom of Tensor Products
			(b) Intersection Alexander–Whitney Maps
			(c) The Diagonal Map
			(d) The Intersection Cup, Cap, and Cross Products
	7.3 Properties of Cup, Cap, and Cross Products.
		7.3.1 Naturality
			(a) Naturality of the Cross Product
			(b) Naturality of Cup and Cap Products
			(c) Compatibility with Classical Products
			(d) Topological Invariance
		7.3.2 Commutativity
		7.3.3 Unitality and Evaluation
			(a) Projection Maps
			(b) Unital Properties of Products
			(c) Products and Evaluations
		7.3.4 Associativity
			(a) Associativity under Broad Assumptions
			(b) Associativity in Some More Specific Settings
		7.3.5 Stability
			(a) Stability of Cap Products
			(b) Algebra of Shifts and Mapping Cones
			(c) Stability of Cross Products and Cup Products
		7.3.6 Criss-Crosses
			(a) The Relation between Cup and Cross Products
			(b) Interchange Identities under Broad Assumptions
			(c) Interchange Identities in Some More Specific Settings
		7.3.7 Locality
		7.3.8 The Cohomology Künneth Theorem
		7.3.9 Summary of Properties
			(a) Naturality
			(b) Associativity
			(c) Commutativity
			(d) Unital properties
			(e) Evaluations
			(f) Stability
			(g) Combinations – properties that involve multiple types of products
			(h) Locality
			(i) Cohomology Künneth Theorem
		7.3.10 Products on ∂-pseudomanifolds
	7.4 Intersection Cohomology with Compact Supports
8. Poincaré Duality
	8.1 Orientations and Fundamental Classes
		8.1.1 Orientation and Fundamental Classes of Manifolds
		8.1.2 Orientation of CS Sets
		8.1.3 Homological Properties of Orientable Pseudomanifolds
			(a) The Orientation Sheaf
			(b) Homological Theorems
				Mayer–Vietoris Reduction Step
				Distinguished Neighborhood Reduction Step
				Proof for PM-Convex Sets
				Proof for Arbitrary
			(c) Useful Corollaries
		8.1.4 Lack of Global Fundamental Classes for Subzero Perversities
		8.1.5 Invariance of Fundamental Classes
			(a) Fundamental Classes under Change of Perversity
			(b) Fundamental Classes under Change of Stratification
		8.1.6 Intersection Homology Factors the Cap Product
			(a) More General Factorizations
		8.1.7 Product Spaces
	8.2 Poincaré Duality
		8.2.1 The Duality Map
		8.2.2 The Poincaré Duality Theorem
			Mayer–Vietoris Step
			Limit Step
			Euclidean Neighborhood Step
		8.2.3 Duality of Torsion-Free Conditions
		8.2.4 Topological Invariance of Poincaré Duality
	8.3 Lefschetz Duality
		8.3.1 Orientations and Fundamental Classes
			(a) Topological Invariance
		8.3.2 Lefschetz Duality
			(a) Topological Invariance
	8.4 The Cup Product and Torsion Pairings
		8.4.1 Some Algebra
			(a) Pairings
			(b) Torsion Submodules and Torsion-Free Quotients
		8.4.2 The Cup Product Pairing
		8.4.3 The Torsion Pairing
			(a) The Components of λ
			(b) Assembling λ
			(c) The Torsion Pairing Made Explicit
			(d) Symmetry and Nonsingularity
			(e) Another Approach to the Torsion Pairing
		8.4.4 Topological Invariance of Pairings
		8.4.5 Image Pairings
			(a) Nondegeneracy
			(b) The Intersection Cohomology Image Pairing
	8.5 The Goresky–MacPherson Intersection Pairing
		8.5.1 The Intersection Pairing on Manifolds
			(a) What Should the Intersection Product Be?
			(b) The PL Intersection Pairing
		8.5.2 The Intersection Pairing on PL Pseudomanifolds
			(a) A Non-GM Intersection Product
			(b) Almost Full Circle
		8.5.3 An Intersection Pairing on Topological Pseudomanifolds and Some Relations of Goresky and MacPherson
9. Witt Spaces and IP Spaces
	9.1 Witt and IP Spaces
		9.1.1 Witt Spaces
			(a) Dependence ofWitt Spaces on Coefficient Choices
		9.1.2 IP Spaces
		9.1.3 Products and Stratification Independence
			(a) Products ofWitt and IP Spaces
			(b) Independence of Stratification of the Witt and IP Conditions.
	9.2 Self-Pairings
	9.3 Witt Signatures
		9.3.1 Definitions and Basic Properties
			(a) Signatures of Matrices and Pairings
			(b) Witt Signatures
				Witt Signatures of Witt Spaces with Boundaries
			(c) Topological Invariance of Witt Signatures
		9.3.2 Properties of Witt Signatures
		9.3.3 Novikov Additivity
		9.3.4 Perverse Signatures
	9.4 L-Classes
		9.4.1 Outline of the Construction of L-Classes (without Proofs)
			(a) Maps to Spheres and Embedded Subspaces
			(b) Cohomotopy
			(c) The L-Classes
			(d) L-Classes on Smooth Manifolds
			(e) L-Classes for Small Degrees
			(f) Characterizing the L-Classes
				PL Trivial Normally Nonsingular Subspaces
				Umkehr Maps
				Axiomatic Characterization
			(g) Some Notation
			(h) The Proofs
		9.4.2 Maps to Spheres and Embedded Subspaces
		9.4.3 Cohomotopy
		9.4.4 The L-Classes
		9.4.5 L-Classes in Small Degrees
			(a) Extending Properties to Small Degrees
		9.4.6 Characterizing the L-Classes
	9.5 A Survey of Pseudomanifold Bordism Theories
		9.5.1 Bordism
			(a) Bordism Groups
			(b) Bordism Homology Theories
		9.5.2 Pseudomanifold Bordism
			(a) All pseudomanifolds
			(b) Mod 2 Euler Spaces
			(c) Q-Witt Spaces
			(d) K-Witt Spaces
			(e) IP bordism
			(f) Other “Witt-type” spaces
			(g) L-Spaces
			(h) Novikov Conjectures
10. Suggestions for Further Reading
	10.1 Background, Foundations, and Next Texts
		10.1.1 Deeper Background
			(a) Sheaf Theory
			(b) Derived Categories and Verdier Duality
	10.2 Bordism
	10.3 Characteristic Classes
	10.4 Intersection Spaces
	10.5 Analytic Approaches to Intersection Cohomology
		10.5.1 L^2-Cohomology
		10.5.2 Perverse Forms
			(a) Chataur–Saralegi–Tanré Theory
	10.6 Stratified Morse Theory
	10.7 Perverse Sheaves and the Decomposition Theorem
	10.8 Hodge Theory
	10.9 Miscellaneous
Appendix A. Algebra
	A.1 Koszul Sign Conventions
		A.1.1 Why Sign?
		A.1.2 Homological versus Cohomological Grading
		A.1.3 The Chain Complex of Maps of Chain Complexes
		A.1.4 Chain Maps and Chain Homotopies
		A.1.5 Consequences
	A.2 Some More Facts about Chain Homotopies
	A.3 Shifts and Mapping Cones
		A.3.1 Shifts
		A.3.2 Algebraic Mapping Cones
	A.4 Projective Modules and Dedekind Domains
		A.4.1 Projective Modules
		A.4.2 Dedekind Domains
	A.5 Linear Algebra of Signatures
		A.5.1 Signatures of Nonsingular Pairings
		A.5.2 Signatures of Orthogonal Sums
		A.5.3 Antisymmetric Pairings
Appendix B. An Introduction to Simplicial and PL Topology
	B.1 Simplicial Complexes and Euclidean Polyhedra
		B.1.1 Simplicial Complexes
		B.1.2 Euclidean Polyhedra
	B.2 PL Spaces and PL Maps
	B.3 Comparing Our Two Notions of PL Spaces
	B.4 PL Subspaces
	B.5 Cones, Joins, and Products of PL Spaces
	B.6 The Eilenberg–Zilber Shuffle Triangulation of Products
		B.6.1 The Definition of the Eilenberg–Zilber Triangulation
		B.6.2 Realization of Partially Ordered Sets
		B.6.3 Products of Partially Ordered Sets and Their Product Triangulations
		B.6.4 Triangulations of Products of Simplicial Complexes and PL Spaces
		B.6.5 The Simplicial Cross Product
References
	1-17
	18-39
	40-60
	61-83
	84-107
	108-130
	131-154
	155-181
	182-205
	206-226
	227-250
	251-254
Glossary of Symbols
Index