## Algebraic & Geometric Topology

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

Takahito Naito

#### Abstract

Let $X$ be a Poincaré duality space, $Y$ a space and $f:X→Y$ 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 $LfY$ 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 $H∗(LfY;ℚ)$ 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
Revised: 10 May 2011
Accepted: 10 July 2011
First available in Project Euclid: 19 December 2017

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

#### 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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