Analytic and geometric study of stratified spaces
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Bibliographic Information
Analytic and geometric study of stratified spaces
（Lecture notes in mathematics, 1768）
Springer, c2001
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references (p. [215]-226) and index
Description and Table of Contents
Description
stratified collectionsofdiffer- inamore intuitive are Expressed terminology, spaces in This characteristic tiable manifoldswhich are an glued together appropriate way. of manifolds" feature becomes in the name or original "complexes [188] apparent of "manifold collections" See. HASSLERWHITNEY fora the by predecessor [189, 11) of stratified WHITNEY's article from the 1947 modern notion a [188] space. year the birth date of abstract of stratifications. Nev- can be as an theory regarded have considered before 1947 which mathematicians nowadays theless, already topics, of stratified like for at theend ofthe are treated within the theory spaces, example to when nineteenth when or century, algebraic geometers began study singularities and of varieties The interest in simplicialcomplexes triangulations algebraic began. of manifolds" in for the introduction of the so-called was reason "complexes [188] bounded submanifold of the observation that the of a some boundary noncompact lower dimensional Euclidean oftenbe in can decomposed locally finitelymany space inmodern manifolds.
This ofviewhasbeentaken point again geometric analysis up and for instance forms the basis of the of a or a concept manifold-with-boundary manifold-with-corners for MELROSE or RENETHOM example [126, 127] 1.1.19). (see noticed in his work of 1955 that his iterated sets of of asmooth singularities [1661 f R' R' manifolds for certain functions and a : -4 are generic comprise mapping collection" inthe ofWHITNEY WHITNEY 1957in "manifold sense proved [190] [1891.
Table of Contents
Introduction Notation 1 Stratified Spaces and Functional Structures 1.1 Decomposed spaces 1.2 Stratifications 1.3 Smooth Structures 1.4 Local Triviality and the Whitney conditions 1.5 The sheaf of Whitney functions 1.6 Rectifiable curves and regularity 1.7 Extension theory for Whitney functions on regular spaces 2 Differential Geometric Objects on Singular Spaces 2.1 Stratified tangent bundles and Whitney's condition (A) 2.2 Derivations and vector fields 2.3 Differential forms and stratified cotangent bundle 2.4 Metrics and length space structures 2.5 Differential operators 2.6 Poisson structures 3 Control Theory 3.1 Tubular neighborhoods 3.2 Cut point distance and maximal tubular neighborhoods 3.3 Curvature moderate submanifolds 3.4 Geometric implications of the Whitney conditions 3.5 Existence and uniqueness theorems 3.6 Tubes and control data 3.7 Controlled vector fields and integrability 3.8 Extension theorems on controlled spaces 3.9 Thom's first isotopy lemma 3.10 Cone spaces 4 Orbit Spaces 4.1 Differentiable G-Manifolds 4.2 Proper Group Actions 4.3 Stratification of the Orbit Space 4.4 Functional Structure 5 DeRham-Cohomology 5.1 The deRham complex on singular spaces 5.2 DeRham cohomology on C^/infty-cone spaces 5.3 DeRham theorems on orbit spaces 5.4 DeRham cohomology of Whitney functions 6 Homology of Algebras of Smooth Functions 6.1 Topological algebras and their modules 6.2 Homological algebra for topological modules 6.3 Continuous Hochschild homology 6.4 Hochschild homology of algebras of smooth functions A Supplements from linear algebra and functional analysis A.1 The vector space distance A.2 Polar decomposition A.3 Topological tensor products B Kahler differentials B.1 The space of Kahler differentials B.2 Topological version B.3 Application to locally ringed spaces C Jets, Whitney functions and a few C^/infty -mappings C.1 Frechet topologies for C^/infty -functions C.2 Jets C.3 Whitney functions C.4 Smoothing of the angle
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