Homotopic invariance of dihedral homologies for 𝐴_{∞}-algebras with involution

Lapin, S.

doi:10.1090/spmj/1736

Homotopic invariance of dihedral homologies for $A_\infty$-algebras with involution
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by S. V. Lapin
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 33 (2022), 949-969

DOI: https://doi.org/10.1090/spmj/1736

Published electronically: October 31, 2022

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Abstract:

It is established that the dihedral homologies of involutive $A_{\infty }$-algebras are homotopically invariant with respect to the homotopy equivalences of involutive $A_{\infty }$-algebras. As a consequence, it is shown that over any field, the dihedral homologies of a topological space are isomorphic to the dihedral homologies of the involutive $A_{\infty }$-algebra of homologies for the simplicial group of Kan loops of the original topological space.

References

S. V. Lapin, Dihedral modules with $\infty$-simplicial faces and dihedral homology of involutive $A_\infty$-algebras over rings, Algebra i Analiz 33 (2021), no. 3, 85–110. (Russian)
S. V. Lapin, Homotopy simplicial faces and the homology of realizations of simplicial topological spaces, Math. Notes 94 (2013), no. 5-6, 619–635. Translation of Mat. Zametki 94 (2013), no. 5, 661–681. MR 3227008, DOI 10.1134/S0001434613110035
S. V. Lapin, Differential perturbations and $D_\infty$-differential modules, Mat. Sb. 192 (2001), no. 11, 55–76 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 11-12, 1639–1659. MR 1886370, DOI 10.1070/SM2001v192n11ABEH000609
R. L. Krasauskas, S. V. Lapin, and Yu. P. Solov′ev, Dihedral homology and cohomology, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1987), 28–32, 102 (Russian). MR 913068
B. L. Tsygan, Homology of certain matrix Lie superalgebras, Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 90–91 (Russian). MR 847158
C. Braun, Involutive $A_{\infty }$-algebras and dihedral cohomology, arXiv:1209.1261v4 (2013).
T. V. Kadeišvili, On the theory of homology of fiber spaces, Uspekhi Mat. Nauk 35 (1980), no. 3(213), 183–188 (Russian). International Topology Conference (Moscow State Univ., Moscow, 1979). MR 580645
R. L. Krasauskas and Yu. P. Solov′ev, Rational Hermitian $K$-theory and dihedral homology, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 935–969, 1118 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 2, 261–293. MR 972090, DOI 10.1070/IM1989v033n02ABEH000826
S. V. Lapin, Differential Lie modules over curved colored coalgebras and $\infty$-simplicial modules, Math. Notes 96 (2014), no. 5-6, 698–715. Translation of Mat. Zametki 96 (2014), no. 5, 709–731. MR 3343633, DOI 10.1134/S0001434614110091
Alain Connes, Cohomologie cyclique et foncteurs $\textrm {Ext}^n$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 953–958 (French, with English summary). MR 777584
S. V. Lapin, Cyclic modules with $\infty$-simplicial faces and the cyclic homology of $A_\infty$-algebras, Mat. Zametki 102 (2017), no. 6, 874–895 (Russian, with Russian summary); English transl., Math. Notes 102 (2017), no. 5-6, 806–823. MR 3733331, DOI 10.4213/mzm11301
Sergey V. Lapin, Cyclic homology of cyclic $\infty$-simplicial modules, Georgian Math. J. 25 (2018), no. 4, 571–591. MR 3881493, DOI 10.1515/gmj-2018-0053
S. V. Lapin, The homotopy invariance of cyclic homology of $A_{\infty }$-algebras over rings, arXiv: 1905.10991 (2019).
V. A. Smirnov, Simplicial and operad methods in algebraic topology, Translations of Mathematical Monographs, vol. 198, American Mathematical Society, Providence, RI, 2001. Translated from the Russian manuscript by G. L. Rybnikov. MR 1811110, DOI 10.1090/mmono/198
James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312. 108 (1963), 275-292; ibid. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
S. V. Lapin, The tensor functor from the category of $A_{\infty }$-algebras into the category of differential modules with $\infty$-simplicial faces, arXiv:1903.00869 (2019).