Access Type
Open Access Dissertation
Date of Award
January 2021
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
John R. Klein
Second Advisor
Vladimir Chernyak
Abstract
An intersection problem consists of submanifolds $P, Q \subset M$ having non-empty intersection. In 1974, Hatcher and Quinn introduced a bordism-theoretic obstruction to finding a deformation of $P$ off of $Q$ by an isotopy. This dissertation studies the problem of finding an analytical expression for the Hatcher-Quinn obstruction---one which involves the language of differential forms. We first introduce the notion of a smooth structure on a set by introducing a system of mappings called plots. By generalizing this to the fibered setting, we use the concept to give a model for the homology of the generalized path space $E$ i.e., the space of paths in $M$ which start at a point of $P$ and end at a point of $Q;$ this is the content of chapter 2. In chapter 3, we introduce diagonal forms and Thom forms. We then use these to construct Hatcher-Quinn forms. Chapter 4 introduces cosimplicial spaces and the geometric cobar construction to give a combinatorial model for the generalized path space. The end of chapter 4 gives a construction of the Hatcher-Quinn functional on the generalized path space. In chapter 5, we develop the bar complex, develop the Hatcher-Quinn functional on the bar complex and show that the functional is co-closed.
Recommended Citation
Turner, Joshua Lenwood, "The Hatcher-Quinn Invariant And Differential Forms" (2021). Wayne State University Dissertations. 3504.
https://digitalcommons.wayne.edu/oa_dissertations/3504