Thom space

Abstract: We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum $R$, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space $X$ to the infinity category of free rank one $R$-modules, which we show is a model for $BGL_1 R$; we show that $BGL_1 R$ classifies homotopy sheaves of rank one $R$-modules, which we call $R$-line bundles. We use our $R$-module Thom spectrum to define the twisted $R$-homology and cohomology of an $R$-line bundle over a space $X$, classified by a map from $X$ to $BGL_1 R$, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. An earlier version of this paper was part of arXiv:0810.4535.

Abstract: We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [17], but our proofs are fundamentally different. Errata Minor errors were corrected on page 967 (18 February 2008).

Abstract: 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 Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\em every} singularity with local algebra Q can be recovered from Ts_Q.

Abstract: The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we can construct Z- and Ngraded algebras, the Z-graded algebra being a Hopf-Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the N-graded algebra. In this case, we can carry out the associated circle bundle and Thom constructions. Starting with a C � algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C � -algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi & Nomizu.