Stable cohomology of graph complexes. (English) Zbl 1518.81051
Summary: We study three graph complexes related to the higher genus Grothendieck-Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus \(g\) tends to \(\infty\). As a byproduct, we find that the Malcev completion of the genus \(g\) mapping class group relative to the symplectic group is Koszul in the stable limit, partially answering a question of Hain.
MSC:
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 18G35 | Chain complexes (category-theoretic aspects), dg categories |
| 18G85 | Graph complexes and graph homology |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
References:
| [1] | Alekseev, A.; Kawazumi, N.; Kuno, Y.; Naef, F., Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra, C. R. Math. Acad. Sci. Paris, 355, 2, 123-127 (2017) · Zbl 1420.57052 · doi:10.1016/j.crma.2016.12.007 |
| [2] | Alekseev, A., Kawazumi, N., Kuno, Y., Naef, F.: The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera (2018). Preprint arXiv:1804.09566 |
| [3] | Alekseev, A.; Torossian, C., The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math. (2), 175, 2, 415-463 (2012) · Zbl 1243.22009 · doi:10.4007/annals.2012.175.2.1 |
| [4] | Asada, M., Kaneko, M.: On the automorphism group of some pro-\(l\) fundamental groups. Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986). Adv. Stud. Pure Math. 12, 137-159, North-Holland, Amsterdam (1987) · Zbl 0657.20028 |
| [5] | Berglund, A., Koszul spaces, Trans. Am. Math. Soc., 366, 9, 4551-4569 (2014) · Zbl 1301.55006 · doi:10.1090/S0002-9947-2014-05935-7 |
| [6] | Berglund, A.; Börjeson, K., Free loop space homology of highly connected manifolds, Forum Math., 29, 1, 201-228 (2017) · Zbl 1422.55023 · doi:10.1515/forum-2015-0074 |
| [7] | Berglund, A.; Madsen, I., Homological stability of diffeomorphism groups, Pure Appl. Math. Q., 9, 1, 1-48 (2013) · Zbl 1295.57038 · doi:10.4310/PAMQ.2013.v9.n1.a1 |
| [8] | Berglund, A.; Madsen, I., Rational homotopy theory of automorphisms of manifolds, Acta Math., 224, 1, 67-185 (2020) · Zbl 1441.57033 · doi:10.4310/ACTA.2020.v224.n1.a2 |
| [9] | Bökstedt, M., Minuz, E.: Graph cohomologies and rational homotopy type of configuration spaces (2019). Preprint. arXiv:1904.01452 |
| [10] | Borel, A., Linear Algebraic Groups. Graduate Texts in Mathematics (1991), New York: Springer, New York · Zbl 0726.20030 · doi:10.1007/978-1-4612-0941-6 |
| [11] | Campos, R., Willwacher, T.: A model for configuration spaces of points. Preprint arXiv:1604.02043 |
| [12] | Chan, M., Galatius, S., Payne, S.: Topology of moduli spaces of tropical curves with marked points (2019). arxiv:1903.07187 |
| [13] | Enriquez, B., Elliptic associators, Selecta Math. (N.S.), 20, 2, 491-584 (2014) · Zbl 1294.17012 · doi:10.1007/s00029-013-0137-3 |
| [14] | Fulton, W.; Harris, J., Representation Theory. A First course. Graduate Texts in Mathematics, Readings in Mathematics (1991), New York: Springer, New York · Zbl 0744.22001 |
| [15] | Felder, M.: Graph complexes and higher genus Grothendieck-Teichmüller Lie algebras (2021). Preprint. arXiv:2105.02056 |
| [16] | Gatsinzi, J-B, The homotopy Lie algebra of classifying spaces, J. Pure Appl. Algebra, 120, 3, 281-289 (1987) · Zbl 0884.55009 · doi:10.1016/S0022-4049(96)00037-0 |
| [17] | Ginzburg, V.; Kapranov, M., Koszul duality for operads, Duke Math. J., 76, 1, 203-272 (1994) · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4 |
| [18] | Hain, R., Infinitesimal presentations of the Torelli groups, J. Am. Math. Soc., 10, 3, 597-651 (1997) · Zbl 0915.57001 · doi:10.1090/S0894-0347-97-00235-X |
| [19] | Hain, R., Genus 3 mapping class groups are not Kähler, J. Topol., 8, 1, 213-246 (2015) · Zbl 1318.14028 · doi:10.1112/jtopol/jtu020 |
| [20] | Idrissi, N., The Lambrechts-Stanley model of configuration spaces, Invent. Math., 216, 1, 1-68 (2019) · Zbl 1422.55031 · doi:10.1007/s00222-018-0842-9 |
| [21] | Kupers, A., Randal-Williams, O.: On the cohomology of Torelli groups (2019). Preprint arXiv:1901.01862 |
| [22] | Kupers, A., Randal-Williams, O.: On diffeomorphisms of even-dimensional discs (2020). Preprint arXiv:2007.13884 |
| [23] | Kupers, A., Randal-Williams, O.: On the Torelli Lie algebra (2021). Preprint arXiv:2106.16010 |
| [24] | Loday, J-L, Cyclic Homology. Appendix E by María O. Ronco. Grundlehren der Mathematischen Wissenschaften (1992), Berlin: Springer, Berlin · Zbl 0780.18009 |
| [25] | Loday, J-L; Vallette, B., Algebraic operads. Grundlehren Math. Wiss. (2012), Heidelberg: Springer, Heidelberg · Zbl 1260.18001 · doi:10.1007/978-3-642-30362-3 |
| [26] | Millès, J., The Koszul complex is the cotangent complex, Int. Math. Res. Not. IMRN, 3, 607-650 (2012) · Zbl 1247.18007 · doi:10.1093/imrn/rnr034 |
| [27] | Milne, JS, Algebraic Groups. The Theory of Group Schemes of Finite Type Over a Field. Cambridge Studies in Advanced Mathematics (2017), Cambridge: Cambridge University Press, Cambridge · Zbl 1390.14004 · doi:10.1017/9781316711736 |
| [28] | Naef, F.; Qin, Y., The Elliptic Kashiwara-Vergne Lie algebra in low weights, J. Lie Theory, 31, 2, 583-598 (2021) · Zbl 1482.17017 |
| [29] | Neisendorfer, J., The rational homotopy groups of complete intersections, Illinois J. Math., 23, 2, 175-182 (1979) · Zbl 0412.55006 · doi:10.1215/ijm/1256048231 |
| [30] | Payne, S., Willwacher, T.: Weight two compactly supported cohomology of moduli spaces of curves. Preprint. arXiv:2110.05711 |
| [31] | Procesi, C., The invariant theory of \(n\times n\) matrices, Adv. Math., 19, 3, 306-381 (1976) · Zbl 0331.15021 · doi:10.1016/0001-8708(76)90027-X |
| [32] | Raphael, E., Schneps, L.: On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra (2017). Preprint arXiv:1706.08299 |
| [33] | Ševera, P.; Willwacher, T., Equivalence of formalities of the little discs operad, Duke Math. J., 160, 1, 175-206 (2011) · Zbl 1241.18008 · doi:10.1215/00127094-1443502 |
| [34] | Vallette, B., Homotopy theory of homotopy algebras, Ann. Inst. Fourier (Grenoble), 70, 2, 683-738 (2020) · Zbl 1508.18019 · doi:10.5802/aif.3322 |
| [35] | Willwacher, T., Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra, Invent. Math., 200, 3, 671-760 (2015) · Zbl 1394.17044 · doi:10.1007/s00222-014-0528-x |
| [36] | Willwacher, T.: On truncated mapping spaces of \(E_n\) operads. arXiv:1911.00709 |
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