Constructing Certain Spectra from Ad Theories

Abstract

In this dissertation some ad theories and their relation to certain spectra are investigated. Quinn defined bordism-type theories, semisimplicial constructions with a geometric realization that is an Ω ­-spectrum. Here we study Laures and McClure’s ad theories, a strengthening of the axioms of the bordism-type theories. Associated to an ad theory is a Quinn spectrum, an Ω­-spectrum that is the geometric realization of a semisiplicial set coming fromthe ad theory.

The goal of the dissertation is to show equivalences between on one hand the Thom spectrum MSO and the Madsen-Tillmann spectra MT+ (d) and on the other hand some constructed Quinn spectra. This is done working through the singular complex of MSO and MT+ (d).

Transversality is important here. The final step of the comparison of spectra is done by comparing the subset of the singular complex that is transversal to the 0- section of the spaces of the spectra MSO and MT+ (d) to the semisimplicial sets coming from the ad theories. For MSO, that this is an equivalence is a simplicial version of part of Rene Thoms proof about the stable homotopy of MSO.

The ad theories are functors from index categories that come from ball complexes, to a target category we construct. This target category is built out of embedded smooth compactmanifolds with corners.