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E-grāmata: Singular Intersection Homology

(Texas Christian University)
  • Formāts: PDF+DRM
  • Sērija : New Mathematical Monographs
  • Izdošanas datums: 24-Sep-2020
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108895361
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Singular Intersection HomologyPalielināt
  • Formāts: PDF+DRM
  • Sērija : New Mathematical Monographs
  • Izdošanas datums: 24-Sep-2020
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108895361
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Intersection homology is a version of homology theory that extends Poincaré duality and its applications to stratified spaces, such as singular varieties. This is the first comprehensive expository book-length introduction to intersection homology from the viewpoint of singular and piecewise-linear chains. Recent breakthroughs have made this approach viable by providing intersection homology and cohomology versions of all the standard tools in the homology tool box, making the subject readily accessible to graduate students and researchers in topology as well as researchers from other fields. This text includes both new research material and new proofs of previously-known results in intersection homology, as well as treatments of many classical topics in algebraic and manifold topology. Written in a detailed but expository style, this book is suitable as an introduction to intersection homology or as a thorough reference.

Recenzijas

' a detailed and meticulous presentation of intersection homology by singular and PL chains.' Daniel Tanré, European Mathematical Society 'Overall, this monograph is a splendid introduction to noncommutative function-theoretic operator theory. Anyone interested in modern operator theory, function theory, and related areas of analysis will find this book a valuable reference.' Jaydeb Sarkar, MathSciNet

Papildus informācija

The first expository book-length introduction to intersection homology from the viewpoint of singular and piecewise linear chains.
Preface xiii
Notations and Conventions xvii
1 Introduction
1(15)
1.1 What Is Intersection Homology?
1(8)
1.2 Simplicial vs. PL vs. Singular
9(1)
1.3 A Note about Sheaves and Their Scarcity
10(1)
1.4 GM vs. Non-GM Intersection Homology
11(2)
1.5 Outline
13(3)
2 Stratified Spaces
16(70)
2.1 First Examples of Stratified Spaces
18(2)
2.2 Filtered and Stratified Spaces
20(8)
2.2.1 Filtered Spaces
20(4)
2.2.2 Stratified Spaces
24(3)
2.2.3 Depth
27(1)
2.3 Locally Cone-like Spaces and CS Sets
28(6)
2.4 Pseudomanifolds
34(4)
2.5 PL Spaces and PL Pseudomanifolds
38(10)
2.5.1 PL Spaces
39(3)
2.5.2 Piecewise Linear and Simplicial Pseudomanifolds
42(6)
2.6 Normal Pseudomanifolds
48(3)
2.7 Pseudomanifolds with Boundaries
51(4)
2.8 Other Species of Stratified Spaces
55(5)
2.8.1 Whitney Stratified Spaces
55(1)
2.8.2 Thorn-Mather Spaces
56(2)
2.8.3 Homotopically Stratified Spaces
58(2)
2.9 Maps of Stratified Spaces
60(3)
2.10 Advanced Topic: Intrinsic Filtrations
63(11)
2.10.1 Intrinsic PL Filtrations
69(5)
2.11 Advanced Topic: Products and Joins
74(12)
2.11.1 Products of Intrinsic Filtrations
82(4)
3 Intersection Homology
86(49)
3.1 Perversities
86(4)
3.1.1 GM Perversities
87(2)
3.1.2 Dual Perversities
89(1)
3.2 Simplicial Intersection Homology
90(17)
3.2.1 First Examples
93(11)
3.2.2 Some Remarks on the Definition
104(3)
3.3 PL Intersection Homology
107(21)
3.3.1 PL Homology
108(7)
3.3.2 A Useful Alternative Characterization of PL Chains
115(5)
3.3.3 PL Intersection Homology
120(1)
3.3.4 The Relation between Simplicial and PL Intersection Homology
121(7)
3.4 Singular Intersection Homology
128(7)
4 Basic Properties of Singular and PL Intersection Homology
135(52)
4.1 Stratified Maps, Homotopies, and Homotopy equivalences
136(7)
4.2 The Cone Formula
143(3)
4.3 Relative Intersection Homology
146(11)
4.3.1 Further Commentary on Subspace Filtrations
150(3)
4.3.2 Stratified Maps Revisited
153(1)
4.3.3 Reduced Intersection Homology and the Relative Cone Formula
154(3)
4.4 Mayer-Vietoris Sequences and Excision
157(30)
4.4.1 PL Excision and Mayer-Vietoris
158(5)
4.4.2 Singular Subdivision, Excision, and Mayer-Vietoris
163(24)
5 Mayer-Vietoris Arguments and Further Properties of Intersection Homology
187(75)
5.1 Mayer-Vietoris Arguments
188(10)
5.1.1 First Applications: High Perversities and Normalization
194(4)
5.2 Cross Products and the Kiinneth Theorem with a Manifold Factor
198(22)
5.2.1 The Singular Chain Cross Product
199(5)
5.2.2 The PL Cross Product
204(5)
5.2.3 Properties of the Cross Product
209(7)
5.2.4 Kiinneth Theorem when One Factor Is a Manifold
216(4)
5.3 Intersection Homology with Coefficients and Universal Coefficient Theorems
220(14)
5.3.1 Definitions of Intersection Homology with Coefficients
220(6)
5.3.2 Universal Coefficient Theorems
226(8)
5.4 Equivalence of PL and Singular Intersection Homology on PL CS Sets
234(7)
5.4.1 Barycentric Subdivisions and Maps from PL Chains to Singular Chains
235(2)
5.4.2 The Isomorphism of PL and Singular Intersection Homology
237(4)
5.5 Topological Invariance
241(16)
5.5.1 Which Perversities Work?
242(2)
5.5.2 The Statement of the Theorem and Some Corollaries
244(5)
5.5.3 Proof of Topological Invariance
249(8)
5.6 Finite Generation
257(5)
6 Non-GM Intersection Homology
262(91)
6.1 Motivation for Non-GM Intersection Homology
262(4)
6.2 Definitions of Non-GM Intersection Homology
266(10)
6.2.1 First Definition of IH*
266(3)
6.2.2 Second Definition of IH*
269(2)
6.2.3 Third Definition of IH*
271(2)
6.2.4 Non-GM Intersection Homology below the Top Perversity
273(1)
6.2.5 A New Cone Formula
274(1)
6.2.6 Relative Non-GM Intersection Homology and the Relative Cone Formula
275(1)
6.3 Properties of IP H* (X; G)
276(26)
6.3.1 Basic Properties
277(14)
6.3.2 Dimensional Homogeneity
291(8)
6.3.3 Local coefficients
299(3)
6.4 A General Kiinneth Theorem
302(35)
6.4.1 A Key Example: the Product of Cones
303(10)
6.4.2 The Kiinneth Theorem
313(6)
6.4.3 A Relative Kiinneth Theorem
319(2)
6.4.4 Applications of the Kiinneth Theorem
321(4)
6.4.5 Some Technical Stuff: the Proof of Lemma 6.4.2
325(12)
6.5 Advanced Topic: Chain Splitting
337(16)
7 Intersection Cohomology and Products
353(145)
7.1 Intersection Cohomology
355(8)
7.2 Cup, Cap, and Cross Products
363(16)
7.2.1 Philosophy
363(6)
7.2.2 Intersection Homology Cup, Cap, and Cross Products
369(10)
7.3 Properties of Cup, Cap, and Cross Products
379(101)
7.3.1 Naturality
381(11)
7.3.2 Commutativity
392(3)
7.3.3 Unitality and Evaluation
395(9)
7.3.4 Associativity
404(10)
7.3.5 Stability
414(18)
7.3.6 Criss-Crosses
432(20)
7.3.7 Locality
452(9)
7.3.8 The Cohomology Kiinneth Theorem
461(4)
7.3.9 Summary of Properties
465(8)
7.3.10 Products on d-pseudomanifolds
473(7)
7.4 Intersection Cohomology with Compact Supports
480(18)
8 Poincare Duality
498(115)
8.1 Orientations and Fundamental Classes
498(38)
8.1.1 Orientation and Fundamental Classes of Manifolds
499(2)
8.1.2 Orientation of CS Sets
501(4)
8.1.3 Homological Properties of Orientable Pseudomanifolds
505(16)
8.1.4 Lack of Global Fundamental Classes for Subzero Perversities
521(2)
8.1.5 Invariance of Fundamental Classes
523(7)
8.1.6 Intersection Homology Factors the Cap Product
530(5)
8.1.7 Product Spaces
535(1)
8.2 Poincare Duality
536(13)
8.2.1 The Duality Map
536(4)
8.2.2 The Poincare Duality Theorem
540(6)
8.2.3 Duality of Torsion-Free Conditions
546(1)
8.2.4 Topological Invariance of Poincare Duality
547(2)
8.3 Lefschetz Duality
549(19)
8.3.1 Orientations and Fundamental Classes
549(9)
8.3.2 Lefschetz Duality
558(10)
8.4 The Cup Product and Torsion Pairings
568(28)
8.4.1 Some Algebra
568(3)
8.4.2 The Cup Product Pairing
571(3)
8.4.3 The Torsion Pairing
574(13)
8.4.4 Topological Invariance of Pairings
587(3)
8.4.5 Image Pairings
590(6)
8.5 The Goresky-MacPherson Intersection Pairing
596(17)
8.5.1 The Intersection Pairing on Manifolds
596(8)
8.5.2 The Intersection Pairing on PL Pseudomanifolds
604(5)
8.5.3 An Intersection Pairing on Topological Pseudomanifolds and Some Relations of Goresky and MacPherson
609(4)
9 Witt Spaces and IP Spaces
613(90)
9.1 Witt and IP Spaces
614(15)
9.1.1 Witt Spaces
614(7)
9.1.2 IP Spaces
621(2)
9.1.3 Products and Stratification Independence
623(6)
9.2 Self-Pairings
629(3)
9.3 Witt Signatures
632(16)
9.3.1 Definitions and Basic Properties
632(6)
9.3.2 Properties of Witt Signatures
638(5)
9.3.3 Novikov Additivity
643(4)
9.3.4 Perverse Signatures
647(1)
9.4 L-Classes
648(41)
9.4.1 Outline of the Construction of L-Classes (without Proofs)
650(10)
9.4.2 Maps to Spheres and Embedded Subspaces
660(6)
9.4.3 Cohomotopy
666(3)
9.4.4 The L-Classes
669(4)
9.4.5 L-Classes in Small Degrees
673(8)
9.4.6 Characterizing the L-Classes
681(8)
9.5 A Survey of Pseudomanifold Bordism Theories
689(14)
9.5.1 Bordism
689(2)
9.5.2 Pseudomanifold Bordism
691(12)
10 Suggestions for Further Reading
703(10)
10.1 Background, Foundations, and Next Texts
703(3)
10.1.1 Deeper Background
705(1)
10.2 Bordism
706(1)
10.3 Characteristic Classes
707(1)
10.4 Intersection Spaces
708(1)
10.5 Analytic Approaches to Intersection Cohomology
708(2)
10.5.1 L2-Cohomology
708(1)
10.5.2 Perverse Forms
709(1)
10.6 Stratified Morse Theory
710(1)
10.7 Perverse Sheaves and the Decomposition Theorem
710(1)
10.8 Hodge Theory
711(1)
10.9 Miscellaneous
712(1)
Appendix A Algebra
713(26)
A.1 Koszul Sign Conventions
713(7)
A.1.1 Why Sign?
713(1)
A.1.2 Homological versus Cohomological Grading
714(1)
A.1.3 The Chain Complex of Maps of Chain Complexes
715(1)
A.1.4 Chain Maps and Chain Homotopies
716(1)
A.1.5 Consequences
717(3)
A.2 Some More Facts about Chain Homotopies
720(3)
A.3 Shifts and Mapping Cones
723(1)
A.3.1 Shifts
723(1)
A.3.2 Algebraic Mapping Cones
723(1)
A.4 Projective Modules and Dedekind Domains
724(5)
A.4.1 Projective Modules
724(3)
A.4.2 Dedekind Domains
727(2)
A.5 Linear Algebra of Signatures
729(10)
A.5.1 Signatures of Nonsingular Pairings
733(2)
A.5.2 Signatures of Orthogonal Sums
735(1)
A.5.3 Antisymmetric Pairings
736(3)
Appendix B An Introduction to Simplicial and PL Topology
739(30)
B.1 Simplicial Complexes and Euclidean Polyhedra
739(5)
B.1.1 Simplicial Complexes
740(2)
B.1.2 Euclidean Polyhedra
742(2)
B.2 PL Spaces and PL Maps
744(6)
B.3 Comparing Our Two Notions of PL Spaces
750(4)
B.4 PL Subspaces
754(1)
B.5 Cones, Joins, and Products of PL Spaces
755(1)
B.6 The Eilenberg-Zilber Shuffle Triangulation of Products
756(13)
B.6.1 The Definition of the Eilenberg-Zilber Triangulation
757(1)
B.6.2 Realization of Partially Ordered Sets
758(2)
B.6.3 Products of Partially Ordered Sets and Their Product Triangulations
760(3)
B.6.4 Triangulations of Products of Simplicial Complexes and PL Spaces
763(2)
B.6.5 The Simplicial Cross Product
765(4)
References 769(12)
Glossary of Symbols 781(6)
Index 787
Greg Friedman is Professor of Mathematics at Texas Christian University. Professor Friedman's primary research is in geometric and algebraic topology with particular emphases on stratified spaces and high-dimensional knot theory. He has given introductory lecture series on intersection homology at the University of Lille and the Fields Institute for Research in Mathematical Sciences. He has received grants from the National Science Foundation and the Simons Foundation.
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