# The Viterbo Transfer as a Map of Spectra and Twisted Chas-Sullivan Products

By Thomas Kragh
PhD Dissertations
September 2007
Abstract:
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 $\Lambda X$. Claude Viterbo constructed a transfer map $(\Lambda j)^! : H^*(\Lambda L) \to H^*(\Lambda N)$. We prove that this transfer map can be realized as a map of Thom spectra $(\Lambda j)_! : (\Lambda N)^{-TN} \to (\Lambda L)^{-TL+\eta}$, where $\eta$ is a virtual bundle defined by the embedding. John D.S. Jones and Ralph L. Cohen proved that the celebrated Chas-Sullivan product for a manifold $N$ can be realized as a product on the Thom spectrum $(\Lambda N)^{-TN}$, turning it into a ring spectrum. We prove a generalized, twisted'' version of this, proving that the target of $(\Lambda j)_!$ is a Chas-Sullivan type ring spectrum. This leads to the natural conjecture that the Viterbo transfer is a ring spectrum homomorphism. We describe partial results on this conjecture.