#### Access Type

Open Access Dissertation

#### Date of Award

January 2011

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

John R. Klein

#### Abstract

In this dissertation, we study the moduli spaces and CW Structures arising from Morse theory.

Suppose M is a smooth manifold and f is a Morse function on it. We consider the negative gradient flow of f. Suppose the flow satisfies transversality. This naturally defines the moduli spaces of flow lines and gives a stratication of M by its unstable manifolds. The gluing of broken flow lines can also be constructed.

We prove that, under certain assumptions, these moduli spaces can be compactified and the compactified spaces are smooth manifolds with corners. Moreover, these compactified manifolds satisfy certain orientation formulas. We also prove that the stratication of M is actually a CW decomposition of M with explicit characteristic maps, which has good properties. Finally, we show that the associativity of gluing of broken flow lines exclusively follows from the compatibility of the manifold structures of the compactified moduli spaces, which establishes the associativity of gluing in certain cases.

In order to obtain the above results, we also prove some results on the dynamical aspects of negative gradient flows, which may be of independent interest.

#### Recommended Citation

Qin, Lizhen, "Moduli spaces and cw structures arising from morse theory" (2011). *Wayne State University Dissertations*. 328.

https://digitalcommons.wayne.edu/oa_dissertations/328