Bott–Thom isomorphism, Hopf bundles and Morse theory

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $$V \rightarrow X$$ V → X of rank divisible by four over a finite complex X we derive a stable decomposition result for vector bundles over the sphere bundle $$\mathord {{\mathbb {S}}}( \mathord {{\mathbb {R}}}\oplus V)$$ S ( R ⊕ V ) in terms of vector bundles and Clifford module bundles over X. After passing to topological K-theory these results imply classical Bott–Thom isomorphism theorems.

A Computation with the Connes–Thom Isomorphism

Let A ∊ Mn(ℝ) be an invertible matrix. Consider the semi-direct product ℝn ⋊ ℤ where the action of ℤ on ℝn is induced by the left multiplication by A. Let (α, τ) be a strongly continuous action of ℝn ⋊ ℤ on a C*-algebra B where α is a strongly continuous action of ℝn and τ is an automorphism. The map τ induces a map . We show that, at the K-theory level, τ commutes with the Connes–Thom map if det(A) > 0 and anticommutes if det(A) < 0. As an application, we recompute the K-groups of the Cuntz–Li algebra associated with an integer dilation matrix.

Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory

We develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

INTERSECTION COHOMOLOGY, MONODROMY AND THE MILNOR FIBER

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus.

Thom isomorphism and push-forward map in twisted K-theory

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). We also obtain the wrong way functoriality property for the push-forward map in twisted K-theory. For D-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.