The Viterbo Transfer as a Map of Spectra

Abstract

Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $j:L\to T^{\ast }N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $\Lambda X$. C. Viterbo constructed a transfer map $(\Lambda j{)}^{!}:H^{\ast }(\Lambda L)\to H^{\ast }(\Lambda N)$. This transfer was constructed using finite dimensional approximation of Floer homology. In this paper we define a family of finite dimensional approximations and realize this transfer as a map of Thom spectra: $(\Lambda j{)}_{!}:(\Lambda N{)}^{-TN}\to (\Lambda L{)}^{-TL+\eta }$, where $\eta$ is a virtual vector bundle classified by the tangential information of $j$.

arXiv comment100 pages, 10 figures. Written in PDFLatex. The third version is an extensive rewrite of the second (including a correction of a small error on the bound of r in the new Proposition 10.1 - this appeared in the second version - but not the first version). The second version is an extensive rewrite of the first