Algebraic & Geometric Topology

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

Takahito Naito

Full-text: Open access


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • J Block, A Lazarev, André–Quillen cohomology and rational homotopy of function spaces, Adv. Math. 193 (2005) 18–39
  • U Buijs, A Murillo, The rational homotopy Lie algebra of function spaces, Comment. Math. Helv. 83 (2008) 723–739
  • M Chas, D Sullivan, String topology, to appear in Ann. of Math. $(2)$
  • K-T Chen, Pullback de Rham cohomology of the free path fibration, Trans. Amer. Math. Soc. 242 (1978) 307–318
  • Y Félix, S Halperin, J-C Thomas, Differential graded algebras in topology, from: “Handbook of algebraic topology”, (I M James, editor), North-Holland, Amsterdam (1995) 829–865
  • Y Félix, S Halperin, J-C Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer, New York (2001)
  • Y Félix, J Oprea, D Tanré, Algebraic models in geometry, Oxford Graduate Texts in Math. 17, Oxford Univ. Press (2008)
  • Y Félix, J-C Thomas, Monoid of self-equivalences and free loop spaces, Proc. Amer. Math. Soc. 132 (2004) 305–312
  • Y Felix, J-C Thomas, M Vigué-Poirrier, The Hochschild cohomology of a closed manifold, Publ. Math. Inst. Hautes Études Sci. (2004) 235–252
  • K Gruher, P Salvatore, Generalized string topology operations, Proc. Lond. Math. Soc. $(3)$ 96 (2008) 78–106
  • S Halperin, J-M Lemaire, Notions of category in differential algebra, from: “Algebraic topology–-rational homotopy (Louvain-la-Neuve, 1986)”, (Y Felix, editor), Lecture Notes in Math. 1318, Springer, Berlin (1988) 138–154
  • K Hess, P-E Parent, J Scott, CoHochschild homology of chain coalgebras, J. Pure Appl. Algebra 213 (2009) 536–556
  • J D S Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987) 403–423
  • G Lupton, S B Smith, Rationalized evaluation subgroups of a map. I: Sullivan models, derivations and $G$–sequences, J. Pure Appl. Algebra 209 (2007) 159–171
  • W S Massey, Singular homology theory, Graduate Texts in Math. 70, Springer, New York (1980)
  • J McCleary, A user's guide to spectral sequences, second edition, Cambridge Studies in Advanced Math. 58, Cambridge Univ. Press (2001)
  • J M Møller, M Raussen, Rational homotopy of spaces of maps into spheres and complex projective spaces, Trans. Amer. Math. Soc. 292 (1985) 721–732
  • P Selick, Introduction to homotopy theory, Fields Institute Monogr. 9, Amer. Math. Soc. (1997)
  • J-P Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. $(2)$ 54 (1951) 425–505
  • M Vigué-Poirrier, Décompositions de l'homologie cyclique des algèbres différentielles graduées commutatives, $K$–Theory 4 (1991) 399–410
  • G W Whitehead, Elements of homotopy theory, Graduate Texts in Math. 61, Springer, New York (1978)