Hochschild cohomology of Sullivan algebras and mapping spaces between manifolds

Document Type : Research Article

Author

Department of Mathematics and Statistical Sciences‎, ‎Botswana International University of Science and Technology.

Abstract

‎Let $e‎: ‎N^n \rightarrow M ^m$ be an embedding of closed‎, ‎oriented manifolds of dimension $n$ and $m$ respectively‎. ‎We study the relationship between the homology of the free loop space $LM$ on $M$ and of the space $L_NM$ of loops of $M$ based in $N$ and define a shriek map‎ ‎$ H_*(e)_{!}‎: ‎H_*( LM‎, ‎\mathbb{Q}) \rightarrow H_*( L_NM‎, ‎\mathbb{Q})$ using Hochschild cohomology and study its properties‎. ‎In particular we extend a result of F'elix on the injectivity of the map induced by $ \aut_1M \rightarrow \map(N‎, ‎M; f ) $ on rational homotopy groups when $M$ and $N$ have the same dimension and $ f‎: ‎N\rightarrow M $ is a map of non zero degree‎.

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References

Block and A. Lazarev, André-Quillen cohomology and rational homotopy of function spaces, Adv. Math. 193 (2005), no. 1, 18--39.
Buijs and A. Murillo, The rational homotopy Lie algebra of function spaces, Comment. Math. Helv. 83 (2008), 4, 723--739.
Chas and D. Sullivan, String topology, preprint math GT/9911159, 1999.
L. Cohen and J. D. S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002), no. 4, 773--798.
Félix, Mapping spaces between manifolds and the evaluation map, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3763--3768.
Félix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, no. 205, Springer-Verlag, New-York, 2001.
Félix, J.-C. Thomas, and M. Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc. (JEMS) 9 (2008), no. 1, 123--156.
Félix, J-C. Thomas, and M. Vigué-Poirrier, The Hochschild cohomology of a closed manifold, Publ. Math. Inst. Hautes Études Sci. 99 (2004) 235--252.
J.-B. Gatsinzi, Derivations, Hochschild cohomology and the Gottlieb group, Homotopy Theory of Function Spaces and Related Topics (Y. Félix, G. Lupton, and S. Smith, eds.), Contemporary Mathematics, vol. 519, American Mathematical Society, Providence, 2010, pp. 93--104.
J.-B Gatsinzi, Hochschild cohomology of a Sullivan algebra, Mediterr. J. Math. 13 (2016), no. 6, 3765--3776.
J.-B. Gatsinzi, Hochschild cohomology of Sullivan algebras and mapping spaces, Arab J. Math. Sci. 25 (2019), no. 1, 123--129.
D. S. Jones, Cyclic homology and equivariant homology, Inv. Math. 87 (1987), no. 2, 403--423.
Lambrechts and D. Stanley, The rational homotopy type of configuration spaces of two points, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 1029--1052.
Lambrechts and D. Stanley, Poincaré duality and commutative differential graded algebras, Ann. Sci. Éc. Norm. Supér. 41 (2008), no. 4, 495--509.
Lupton and S.B. Smith, Rationalized evaluation subgroups of a map I: Sullivan models, derivations and G-sequences, J. Pure Appl. Algebra 209 (2007), no. 1, 159--171.
Sulliivan, Open and closed string field theory interpreted in classical algebraic topology, Topology, Geometry and Quantum Field Theory, London Math.Soc. Lecture Notes, vol. 308, Cambridge University Press, 2004, pp. 344--357.
Sullivan, Infinitesimal computations in topology, Inst. Hautes tudes Sci. Publ. Math. 47 (1977) 269--331.
Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geom. 11 (1976), no. 4, 633--644.
Journal of the Iranian Mathematical Society
Volume 3, Issue 1
January 2022
Pages 23-32

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