In this work we construct from ground up a homotopy theory of C ∗-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C ∗-algebras. The spaces in C ∗-homotopy theory are certain hybrids of functors represented by C ∗-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C ∗-algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C ∗-homotopy theory. The stable homotopy category of C ∗-algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We

and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.

Abstract not found

Abstract not found