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Koszul A∞-algebras and free loop space homology

Part of: Rings and algebras arising under various constructions Homotopy theory

Published online by Cambridge University Press:  18 July 2019

Alexander Berglund  and
Kaj Börjeson
Alexander Berglund
Affiliation:
Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden ([email protected])
Kaj Börjeson
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark ([email protected])
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Abstract

We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Keywords

Koszul dualityloop spacesA∞ algebras

MSC classification

Primary: 55P50: String topology 16S37: Quadratic and Koszul algebras
Secondary: 55P35: Loop spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 37 - 65
Copyright
Copyright © Edinburgh Mathematical Society 2019

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References

1.Berglund, A., Homological perturbation theory for algebras over operads, Algebr. Geom. Topol. 14(5) (2014), 2511–2548.CrossRefGoogle Scholar
2.Berglund, A., Koszul spaces, Trans. Amer. Math. Soc. 366(9) (2014), 4551–4569.CrossRefGoogle Scholar
3.Berglund, A. and Börjeson, K., Free loop space homology of highly connected manifolds, Forum Math. 29(1) (2017), 201–228.CrossRefGoogle Scholar
4.Chas, M. and Sullivan, D., String topology (arXiv: math/9911159, 1999).Google Scholar
5.Cohen, R. L. and Jones, J. D. S., A homotopy theoretic realization of string topology, Math. Ann. 324(4) (2002), 773–798.CrossRefGoogle Scholar
6.Cohen, R. L., Jones, J. D. S. and Yan, J., The loop homology algebra of spheres and projective spaces, in Categorical decomposition techniques in algebraic topology (Isle of Skye) 2001 (eds. Arone, G., Hubbuck, J., Levi, R. and Weiss, M.), Progress in Mathematics, Volume 215, pp. 77–92 (Birkhäuser, Basel, 2004).Google Scholar
7.Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds, Invent. Math. 29(3) (1975), 245–274.CrossRefGoogle Scholar
8.Dotsenko, V. and Vallette, B., Higher Koszul duality for associative algebras, Glasg. Math. J. 55(A) (2013), 55–74.CrossRefGoogle Scholar
9.Félix, Y., Halperin, S. and Thomas, J.-C., Differential graded algebras in topology, Handbook of Algebraic Topology, pp. 829–865 (North-Holland, Amsterdam, 1995).Google Scholar
10.Félix, Y., Menichi, L. and Thomas, J.-C., Gerstenhaber duality in Hochschild cohomology, J. Pure Appl. Algebra 199(1–3) (2005), 43–59.CrossRefGoogle Scholar
11.Félix, Y., Oprea, J. and Tanré, D., Algebraic models in geometry, Oxford Graduate Texts in Mathematics, Volume 17 (Oxford University Press, Oxford, 2008).Google Scholar
12.Godin, V., Higher string topology operations (arXiv: 0711.4859v2, 2008).Google Scholar
13.Goresky, M. and Hingston, N., Loop products and closed geodesics, Duke Math. J. 150(1) (2009), 117–209.CrossRefGoogle Scholar
14.Halperin, S. and Stasheff, J., Obstructions to homotopy equivalences, Adv. in Math. 32(3) (1979), 233–279.CrossRefGoogle Scholar
15.He, J.-W. and Lu, D.-M., Higher Koszul algebras and A-infinity algebras, J. Algebra 293(2) (2005), 335–362.CrossRefGoogle Scholar
16.Keller, B., Introduction to A-infinity algebras and modules, Homology Homotopy Appl. 3(1) (2001), 1–35.CrossRefGoogle Scholar
17.Lefèvre-Hasegawa, K., Sur les A-infini catégories (arXiv: math/0310337v1, 2003).Google Scholar
18.Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, Volume 346 (Springer, Heidelberg, 2012).CrossRefGoogle Scholar
19.Malm, E., String topology and the based loop space, PhD thesis, Stanford University (2010).Google Scholar
20.Millès, J., The Koszul complex is the cotangent complex, Int. Math. Res. Not. IMRN 3 (2012), 607–650.CrossRefGoogle Scholar
21.Polishchuk, A. and Positselski, L., Quadratic algebras, University Lecture Series, Volume 37 (American Mathematical Society, Providence, RI, 2005).Google Scholar
22.Priddy, S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60.CrossRefGoogle Scholar
23.Prouté, A., A ∞-structures. Modèles minimaux de Baues-Lemaire et Kadeishvili et homologie des fibrations, Repr. Theory Appl. Categ. 21 (2011), 1–99.Google Scholar
24.Saleh, B., Noncommutative formality implies commutative and Lie formality, Algebr. Geom. Topol. 17(4) (2017), 2523–2542.CrossRefGoogle Scholar
25.Stasheff, J. D., Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275–292; ibid. 108 (1963), 293–312.Google Scholar