Comparison of Cobordism Theories

Motivic Strict Ring Spectra Representing Semi-topological Cohomology Theories

We show that Shipley’s “detection functor” for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological K-theory, and semitopological cobordism for complex varieties. As a further application to semi-topological cobordism, we show that it is related to semi-topological K-theory via a...

Stable Algebraic Topology, 1945–1966

1. Setting up the foundations 3 2. The Eilenberg-Steenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 5. Steenrod operations, K(π, n)’s, and characteristic classes 8 6. The introduction of cobordism 10 7. The route from cobordism towards K-theory 12 8. Bott periodicity and K-theory 14 9. The Adams spectral sequence and Hopf...

Pirates of the Cobordism: Curse of the Knot Surgery

Two n-dimensional manifolds M1 and M2 are cobordant if there is an (n + 1)-dimensional manifold W with the disjoint union of M1 and M2 as its boundary. Such a simple definition can lead to very interesting results in many different areas of mathematics. Algebraic topology is perhaps the most obvious one, with cobordism being central to surgery theory and the study of high-dimensional manifolds....

Topological Quantum Field Theories on Generalized Cobordism Categories

Notes on Spin manifolds with singularitiesy

We study cobordism theories of Spin manifolds with particular singu-larities. The objective here is to compute coeecient groups and 2-local homotopy type of the corresponding classifying spectra in classical terms of cobordism theory, in particular, in terms of the Conner-Floyd characteristic numbers in the corresponding K-theory.