#### 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