On the mapping space homotopy groups and the free loop space homology groups

Let $X$ be a Poincaré duality space, $Y$ a space and $f : X \to Y$ a based map. We show that the rational homotopy group of the connected component of the space of maps from $X$ to $Y$ containing $f$ is contained in the rational homology group of a space $L_{f} Y$ which is the pullback of $f$ and the evaluation map from the free loop space $L Y$ to the space $Y$ . As an application of the result, when $X$ is a closed oriented manifold, we give a condition of a noncommutativity for the rational loop homology algebra $H_{*} (L_{f} Y; ℚ)$ defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.

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Keywords

string topology, Hochschild (co)homology, mapping space, free loop space, rational homotopy theory

Mathematical Subject Classification 2010

Primary: 55P35, 55P50

Secondary: 55P62

References

Publication

Received: 26 January 2011

Revised: 10 May 2011

Accepted: 10 July 2011

Published: 5 September 2011

Authors

Takahito Naito
Interdisciplinary Graduate School of Science and Technology Shinshu University 3-1-1 Asahi Matsumoto, Nagano 390-8621 Japan

Volume 11, issue 4 (2011)

Takahito Naito

Abstract