A semi-infinite cycle construction of Floer homology

This dissertation is concerned with the foundations of a new approach to Floer theory. As opposed to the traditional approach, which can be viewed as a generalization of Morse theory to an infinite dimensional setting, our approach is a generalization of bordism to infinite dimensions. The key new insight, based on unpublished work of Tom Mrowka and Peter Ozsvath, is an understanding of how to axiomatize compactness in the infinite dimensional setting. We describe a general axiomatic framework for setting up a Floer theory of a polarized Hilbert space equipped with a functional. The resulting bordism theory can be seen as a refinement of the traditional Floer theory. By introducing cycles with corners, we demonstrate how the bordism theory leads to a geometric description of homology. We relate our geometric construction to the Morse-theoretic approach by indicating how one might compute the Floer homology of the space, if the associated functional is Morse. The general theory is illustrated in two examples: Seiberg-Witten-Floer homology and symplectic Floer theory for loops in Cn. We end by indicating various generalizations of the theory.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2008

Author dc:creator

• Lipyanskiy, Maksim

Advisor dc:contributor.advisor

• Tomasz Mrowka.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/43785

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/43785