We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f : X → BF , where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild h...

The Chern classes for degeneracy loci of quivers are natural generalizations of the Thom-Porteous-Giambelli formula. Suppose that E, F are vector bundles over a manifold M and that s : E → F is a vector bundle homomorphism. The question is, which cohomology class is defined by the set 6k(s) ⊂ M consisting of points m where the linear map s(m) has corank k? The answer, due to I. Porteous, is a d...

Using the Mathai-Quillen formalism we reexamine the twisted N = 4 supersymmetric model of Vafa-Witten theory. Smooth out the relation between the supersymmetric action and the path integral representation of the Thom class.

Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the T...

It is no accident that Peter Landweber’s career closely matches the striking unification of algebra and topology provided by the theory of bordism of manifolds, especially complex bordism: his work has been at the heart of that interaction. Here I will briefly describe some of Landweber’s main contributions to this story. The starting point was Thom’s work [59], from 1954, in which he used tran...

We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T-spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th ∈ MSp4,2(MSp2), a tautological Pontryagin class p MSp 1 ∈ MSp (HP) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove ...

We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of theWitten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra.

Introduction 2 1. Thom Spaces 3 1.1. G-framed submanifolds. Classes of L-equivalent submanifolds 3 1.2. Thom spaces. The classifying properties of Thom spaces 4 1.3. The cohomologies of Thom spaces modulo p for p > 2 6 1.4. Cohomologies of Thom spaces modulo 2 8 1.5. Diagonal Homomorphisms 11 2. Inner Homology Rings 13 2.1. Modules with One Generator 13 2.2. Modules over the Steenrod Algebra. T...

1. Smooth Manifolds: 4/7/15 1 2. The Tangent Bundle: 4/9/15 4 3. Parallelizability: 4/14/15 6 4. The Whitney Embedding Theorem: 4/16/15 8 5. Immersions, Submersions and Tubular Neighborhoods: 4/21/15 10 6. Cobordism: 4/23/15 12 7. Homotopy Theory: 4/28/15 14 8. The Pontryagin-Thom Theorem: 4/30/15 17 9. Differential Forms: 5/5/15 18 10. de Rham Cohomology, Integration of Differential Forms, and...