The book provides an introduction to stratification theory leading the reader up to modern research topics in the field. The first part presents the basics of stratification theory, in particular the Whitney conditions and Mather's control theory, and introduces the notion of a smooth structure. Moreover, it explains how one can use smooth structures to transfer differential geometric and analytic methods from the arena of manifolds to stratified spaces. In the second part the methods established in the first part are applied to particular classes of stratified spaces like for example orbit spaces. Then a new de Rham theory for stratified spaces is established and finally the Hochschild (co)homology theory of smooth functions on certain classes of stratified spaces is studied. The book should be accessible to readers acquainted with the basics of topology, analysis and differential geometry. TOC:Introduction.- Notation.- Stratified Spaces and Functional Structures.- Differential Geometric Objects on Singular Spaces.- Control Theory.- Orbit Spaces.- DeRham-Cohomology.- Homology of Algebras of Smooth Functions.- A Supplements from linear algebra and functional analysis.- B Kähler differentials.- c Jets, Whitney functions and a few C^/infty-mappings.-

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 Kähler differentials B.1 The space of Kähler 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