Continuous homotopy invariance of bivariant local cyclic homology for σ-C^*-algebras

We establish the continuous homotopy invariance of bivariant local cyclic homology on the category of all \sigma-C^*-algebras. The argument relies vitally on an isomorphism between the smooth and continuous cylinder constructions using a technical criterion due to Meyer. As a consequence we compute the local cyclic homology of the infinite sphere.