Algebraic & Geometric Topology

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

Takahito Naito

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Abstract

Let X be a Poincaré duality space, Y a space and f:XY 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 LfY which is the pullback of f and the evaluation map from the free loop space LY 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(LfY;) defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 4 (2011), 2369-2390.

Dates
Received: 26 January 2011
Revised: 10 May 2011
Accepted: 10 July 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715272

Digital Object Identifier
doi:10.2140/agt.2011.11.2369

Mathematical Reviews number (MathSciNet)
MR2835233

Zentralblatt MATH identifier
1237.55006

Subjects
Primary: 55P35: Loop spaces 55P50: String topology
Secondary: 55P62: Rational homotopy theory

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

Citation

Naito, Takahito. On the mapping space homotopy groups and the free loop space homology groups. Algebr. Geom. Topol. 11 (2011), no. 4, 2369--2390. doi:10.2140/agt.2011.11.2369. https://projecteuclid.org/euclid.agt/1513715272


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