The Viterbo transfer as a map of spectra

Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $j : L \to T^{*} N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $\Lambda X$. In “Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces” [J. Differential Geom. 47 (1997), no. 3, 420–468], C. Viterbo constructed a transfer map ${(\Lambda j)}^{!} : H^{*} (\Lambda L) \to H^{*} (\Lambda N)$. 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: ${(\Lambda j)}_{!}: {(\Lambda N)}^{-TN} \to {(\Lambda L)}^{-TL + \eta}$, where $\eta$ is a virtual vector bundle classified by the tangential information of $j$.

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The author was partially funded by CTQM, OP-ALG-TOP-GEO, Topology in Norway, Aarhus University, Carlsberg, M.I.T., Oslo University, and Uppsala University.

Received 22 February 2012

Accepted 31 October 2017

Published 20 April 2018