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Keywords:
Operads and props; $E_\infty$-structures; simplicial sets; diagonal approximations; arc surfaces
Operads and props; $E_\infty$-structures; simplicial sets; diagonal approximations; arc surfaces
Summary:
We construct, using finitely many generating cells and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_\infty$-operad. We use one of these to construct a cellular $E_\infty$-bialgebra structure on the interval and derive from it a natural cellular $E_\infty$-coalgebra structure on the geometric realization of a simplicial set which, passing to cellular chains, recovers up to signs the Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse and McClure-Smith. We use another prop, a quotient of the first, to relate our constructions to earlier work of Kaufmann and prove a conjecture of his. This is the second of two papers in a series, the first investigates analogous constructions in the category of chain complexes.
References:
[1] Berger, Clemens, Fresse, Benoit: Combinatorial operad actions on cochains. Mathematical Proceedings of the Cambridge Philosophical Society, 137(1):135–174 MR 2075046
[2] Boardman, John Michael, Vogt, Rainer M: Homotopy invariant algebraic structures on topological spaces, volume 347. Springer MR 0420609
[3] Brumfiel, Greg, Medina-Mardones, Anibal M., Morgan, John: A cochain level proof of Adem relations in the mod 2 Steenrod algebra. Arxiv:2006.09354 http://arxiv.org/pdf/2006.09354 MR 4470387
[4] Carlsson, Gunnar: Topology and data. Bull. Amer. Math. Soc. (N.S.), 46(2):255–308 MR 2476414
[5] Chas, Moira, Sullivan, Dennis: String topology. Arxiv:math/9911159 http://arxiv.org/pdf/math/9911159
[6] Dotsenko, Vladimir, Khoroshkin, Anton: Gröbner bases for operads. Duke Mathematical Journal, 153(2):363–396 MR 2667136
[7] Fresse, Benoit: Props in model categories and homotopy invariance of structures. Georgian Mathematical Journal, 17(1):79–160 MR 2640648
[8] Goresky, Mark, Hingston, Nancy: Loop products and closed geodesics. Duke Math. J., 150(1):117–209 MR 2560110
[9] Kaufmann, Ralph M.: Moduli space actions on the Hochschild co-chains of a Frobenius algebra. II. Correlators. J. Noncommut. Geom., 2(3):283–332 MR 2411420
[10] Kaufmann, Ralph M.: Noncommutative aspects of open/closed strings via foliations. Rep. Math. Phys., 61(2):281–293 MR 2424095
[11] Kaufmann, Ralph M.: Dimension vs. genus: a surface realization of the little k-cubes and an E_(∞)-operad. In Algebraic topology—old and new, volume 85 of Banach Center Publ., pages 241–274. Polish Acad. Sci. Inst. Math., Warsaw MR 2503531
[12] Kaufmann, Ralph M: A detailed look on actions on Hochschild complexes especially the degree 1 co-product and actions on loop spaces. Arxiv:1807.10534 http://arxiv.org/pdf/1807.10534 MR 4478268
[13] Kaufmann, Ralph M., Livernet, Muriel, Penner, R. C.: Arc operads and arc algebras. Geom. Topol., 7:511–568 MR 2026541
[14] Kaufmann, Ralph M, Medina-Mardones, Anibal M.: Cochain level May-Steenrod operations. Arxiv:2010.02571 http://arxiv.org/pdf/2010.02571 MR 4333989
[15] Kaufmann, Ralph M, Medina-Mardones, Anibal M.: A combinatorial E_(∞) algebra structure on cubical cochains. Arxiv:2107.00669 http://arxiv.org/pdf/2107.00669
[16] Kaufmann, Ralph M., Zhang, Yongheng: Permutohedral structures on E₂-operads. Forum Math., 29(6):1371–1411 MR 3719307
[17] Loday, Jean-Louis, Vallette, Bruno: Algebraic operads, volume 346 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg MR 2954392
[18] May, J Peter: The geometry of iterated loop spaces, volume 271. Springer MR 0420610
[19] McClure, James, Smith, Jeffrey: Multivariable cochain operations and little n-cubes. Journal of the American Mathematical Society, 16(3):681–704 MR 1969208
[20] Medina-Mardones, Anibal M.: An axiomatic characterization of Steenrod’s cup-i products. Arxiv:1810.06505 http://arxiv.org/pdf/1810.06505
[21] Medina-Mardones, Anibal M.: Persistence Steenrod modules. Arxiv:1812.05031 http://arxiv.org/pdf/1812.05031
[22] Medina-Mardones, Anibal M.: An algebraic representation of globular sets. Homology Homotopy Appl., 22(2):135–150 MR 4093174
[23] Medina-Mardones, Anibal M.: An effective proof of the Cartan formula: the even prime. J. Pure Appl. Algebra, 224(12):106444, 18 MR 4102178
[24] Medina-Mardones, Anibal M.: A finitely presented E_(∞)-prop I: algebraic context. High. Struct., 4(2):1–21 MR 4133162
[25] Medina-Mardones, Anibal M.: A computer algebra system for the study of commutativity up-to-coherent homotopies. Arxiv:2102.07670 http://arxiv.org/pdf/2102.07670 MR 4425165
[26] Medina-Mardones, Anibal M.: New formulas for cup-i products and fast computation of Steenrod quares. Arxiv:2105.08025 http://arxiv.org/pdf/2105.08025 MR 4473678
[27] Steenrod, Norman E: Products of cocycles and extensions of mappings. Annals of Mathematics, pages 290–320
[28] Street, Ross: The algebra of oriented simplexes. Journal of Pure and Applied Algebra, 49(3):283–335
[29] Tauzin, Guillaume, Lupo, Umberto, Tunstall, Lewis, Pérez, Julian Burella, Caorsi, Matteo, Medina-Mardones, Anibal M., Dassatti, Alberto, Hess, Kathryn: giotto-tda: A topological data analysis toolkit for machine learning and data exploration. arXiv:2004.02551 MR 4253732
[30] Tradler, Thomas, Zeinalian, Mahmoud: Algebraic string operations. K-Theory, 38(1):59–82 MR 2353864
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