Algebraic Hopf invariants and rational models for mapping spaces

  • Felix WierstraEmail author


The main goal of this paper is to define an invariant \(mc_{\infty }(f)\) of homotopy classes of maps \(f:X \rightarrow Y_{\mathbb {Q}}\), from a finite CW-complex X to a rational space \(Y_{\mathbb {Q}}\). We prove that this invariant is complete, i.e. \(mc_{\infty }(f)=mc_{\infty }(g)\) if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure on certain convolution algebras. More precisely, given an operadic twisting morphism from a cooperad \(\mathcal {C}\) to an operad \(\mathcal {P}\), a \(\mathcal {C}\)-coalgebra C and a \(\mathcal {P}\)-algebra A, then there exists a natural homotopy Lie algebra structure on \(Hom_\mathbb {K}(C,A)\), the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to model mapping spaces. More precisely, suppose that C is a \(C_\infty \)-coalgebra model for a simply-connected finite CW-complex X and A an \(L_\infty \)-algebra model for a simply-connected rational space \(Y_{\mathbb {Q}}\) of finite \(\mathbb {Q}\)-type, then \(Hom_\mathbb {K}(C,A)\), the space of linear maps from C to A, can be equipped with an \(L_\infty \)-structure such that it becomes a rational model for the based mapping space \(Map_*(X,Y_\mathbb {Q})\).


Hopf invariants Rational homotopy theory Homotopy Lie convolution algebra 



The author would like to thank Alexander Berglund and Dev Sinha for many useful conversations and comments on this paper. The author would also like to thank Philip Hackney for answering many of his questions. The author also acknowledges the financial support from Grant GA CR No. P201/12/G028.


  1. 1.
    Berger, C., Moerdijk, I.: Axiomatic homotopy theory for operads. Comment. Math. Helv. 78(4), 805–831 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berglund, A.: Homological perturbation theory for algebras over operads. Algebr. Geom. Topol. 14(5), 2511–2548 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berglund, A.: Koszul spaces. Trans. Am. Math. Soc. 366(9), 4551–4569 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berglund, A.: Rational homotopy theory of mapping spaces via Lie theory for \(L_\infty \)-algebras. Homol. Homotopy Appl. 17(2), 343–369 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bott, R., Loring, W.T.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  6. 6.
    Buijs, U., Gutiérrez, J.J.: Homotopy transfer and rational models for mapping spaces. J. Homotopy Relat. Struct. 11(2), 309–332 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buijs, U., Murillo, A.: Algebraic models of non-connected spaces and homotopy theory of \(L_\infty \) algebras. Adv. Math. 236, 60–91 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dolgushev, V.A., Hoffnung, A.E., Rogers, C.L.: What do homotopy algebras form? Adv. Math. 274, 562–605 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dotsenko, V., Poncin, N.: A tale of three homotopies. Appl. Categ. Struct. 24(6), 845–873 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Drummond-Cole, G.C., Hirsh, J.: Model structures for coalgebras. Proc. Amer. Math. Soc. 144(4), 1467–1481 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dwyer, W.G., Spaliński, J.: Homotopy theories and model categories. In: Handbook of Algebraic Topology, pp. 73–126. North-Holland, Amsterdam (1995)Google Scholar
  12. 12.
    Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)CrossRefGoogle Scholar
  13. 13.
    Fresse, B.: Modules over Operads and Functors, Lecture Notes in Mathematics, vol. 1967. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Getzler, E.: Lie theory for nilpotent \(L_\infty \)-algebras. Ann. Math. (2) 170(1), 271–301 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hinich, V.: Homological algebra of homotopy algebras. Commun. Algebra 25(10), 3291–3323 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Loday, J.-L., Vallette, B.: Algebraic Operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346. Springer, Heidelberg (2012)Google Scholar
  17. 17.
    Quillen, D.: Rational homotopy theory. Ann. Math. 2(90), 205–295 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Robert-Nicoud, D., Wierstra, F.: Convolution algebras and the deformation theory of infinity-morphisms. Homology Homotopy Appl. 21(1), 351–373 (2019)CrossRefzbMATHGoogle Scholar
  19. 19.
    Robert-Nicoud, D., Wierstra, F.: Homotopy morphisms between convolution homotopy lie algebras. J. Noncommutative Geom. (to appear). arXiv:1712.00794v1
  20. 20.
    Sinha, D., Walter, B.: Lie coalgebras and rational homotopy theory II: Hopf invariants. Trans. Am. Math. Soc. 365(2), 861–883 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vallette, B.: Homotopy theory of homotopy algebras. arXiv:1411.5533v3
  22. 22.
    Wierstra, F.: Hopf invariants and differential forms. High. Struct. (to appear). arXiv:1711.04565v1

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© Tbilisi Centre for Mathematical Sciences 2019

Authors and Affiliations

  1. 1.Mathematical Institute of Charles UniversityCharles UniversityPraha 8Czech Republic

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