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

Naito, Takahito

doi:10.2140/agt.2011.11.2369

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

Takahito Naito

Algebraic & Geometric Topology 11 (2011) 2369–2390

DOI: 10.2140/agt.2011.11.2369

Abstract

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.

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)