Semi-Infinite Cycles in Floer Theory: Viterbo's Theorem

TL;DR

This paper develops an axiomatic, geometric foundation for Floer theory using semi-infinite cycles, introduces a bordism version for cotangent bundles, and proves a bordism version of Viterbo's theorem linking Floer bordism to loop space bordism.

Contribution

It provides a new axiomatic and geometric approach to Floer theory that avoids traditional compactness and gluing assumptions, and establishes a bordism version of Viterbo's theorem.

Findings

Introduces semi-infinite cycles in Floer theory.

Defines a bordism version of Floer theory for cotangent bundles.

Proves a bordism version of Viterbo's theorem connecting Floer and loop space bordism groups.

Abstract

This is the first of a series of papers on foundations of Floer theory. We give an axiomatic treatment of the geometric notion of a semi-infinite cycle. Using this notion, we introduce a bordism version of Floer theory for the cotangent bundle of a compact manifold M. Our construction is geometric and does not require the compactness and gluing results traditionally used to setup Floer theory. Finally, we prove a bordism version of Viterbo's theorem relating Floer bordism of the cotangent bundle to the ordinary bordism groups of the free loop space of M.