The Viterbo Transfer as a Map of Spectra

TL;DR

This paper develops a new finite dimensional approximation framework to realize Viterbo's transfer map between free loop space cohomologies as a Thom spectrum map, enriching the understanding of symplectic topology.

Contribution

It introduces a family of finite dimensional approximations to Viterbo's transfer, representing it as a Thom spectrum map involving virtual vector bundles.

Findings

Realization of Viterbo transfer as Thom spectrum map

New finite dimensional approximation methods for Floer homology

Enhanced understanding of symplectic transfer maps

Abstract

Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $j\colon L\to T^*N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $\Lambda X$. C. Viterbo constructed a transfer map $(\Lambda j)^! \colon H^*(\Lambda L) \to H^*(\Lambda N)$. 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: $(\Lambda j)_! \colon (\Lambda N)^{-TN} \to (\Lambda L)^{-TL+\eta}$, where $\eta$ is a virtual vector bundle classified by the tangential information of $j$.