A ug 2 00 2 K - theory of stratified vector bundles Hans - Joachim Baues and Davide L . Ferrario

We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold is such a stratified vector bundle. 1 Families of vector spaces Let R be the field of real numbers and R = R ⊕ · · · ⊕ R be the standard n-dimensional R-vector space. Let Vect denote the category of finite dimensional R-vector spaces and linear maps and V a subcategory of Vect, termed the structure category. For example let V be the subcategory of the surjective maps in Vect. Or let G be a subgroup of the automorphism group GLn(R) of R . Then G yields the subcategory G ⊂ Vect consisting of one object R and morphisms given by elements in G. The category Vect is a topological category. We say that V is a closed subcategory if for objects V , W ∈ V the inclusion of morphisms sets homV(V,W ) ⊂ homVect(V,W ) is a closed subspace. Here homVect(V,W ) is homeomorphic to R N while homV(V,W ) needs not to be a vector space. Following Atiyah [1] a family of vector spaces in V is a topological space X together with a map pX : X → X̄ and for every b ∈ X̄ a homeomorphism, named chart, Φb : p −1 X (b) ≈ Xb where Xb is a suitable object of V depending upon b ∈ X̄. We denote the family (pX : X → X̄,Xb,Φb, b ∈ X̄) simply by X (it will be clear from the context whether X is the underlying space or the family). The map pX is termed the projection, the space X the total space of the family of vector spaces, the space X̄ the base space, and for every b the vector space Xb is termed fiber over b. A family X is also termed V-family, to make explicit the choice of V. The V-family X is discrete if X̄