×

Motivic rational homotopy type. (English) Zbl 07308121

Summary: We propose a motivic generalization of rational homotopy types. The algebraic invariants we study are defined as algebra objects in the category of mixed motives. This invariant plays a role of Sullivan’s polynomial de Rham algebras. Another main notion is that of cotangent motives. Our main objective is to investigate the topological realization of these invariants and study their structures. Applying these machineries and the Tannakian theory, we construct actions of a derived motivic Galois group on rational homotopy types. Thanks to this, we deduce actions of the motivic Galois group of pro-unipotent completions of homotopy groups.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
55P62 Rational homotopy theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
PDFBibTeX XMLCite
Full Text: arXiv

References:

[1] L. Ahlfors and L. Sario, Riemann Surfaces, Princeton Univ. Press, 1960. · Zbl 0196.33801
[2] G. Ancona, S. Enright-Ward and A. Huber, On the motives of a commutative algebraic group, Documenta Math.20(2015), 807-858. · Zbl 1342.14009
[3] Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, periods), Panoramas et Synthèses (Panoramas and Synthèses), vol.17, Société Mathématique de France, Paris, 2004. · Zbl 1060.14001
[4] Y. André and B. Kahn, Nilpotence, radicaux et structures monoidales, Rend. Sem. Math. Univ. Padova,108, (2002), 107-291. · Zbl 1165.18300
[5] C. Berger and B. Fresse, Combinatorial operad operations of cochains, Math. Proc. Cambridge Phil. Soc.137(2004), 135-174 · Zbl 1056.55006
[6] J. Bergner, A survey of(∞,1)-categories,Towards higher categories, IMA Volumes in Mathematics and Its Applications 152, Springer 69-83, 2010. · Zbl 1200.18011
[7] D. Ben-Zvi and D. Nadler, Loop space and connections, J. Topol.5(2012) 377-430. · Zbl 1246.14027
[8] A. Borel, Linear Algebraic Groups, (the second edition) Springer-Verlag 1991. · Zbl 0726.20030
[9] R. Bott and L.W. Tu, Differential forms in Algebraic Topology, Graduate texts in mathematics vol. 82 Springer, 1982. · Zbl 0496.55001
[10] J. Carlson, H. Clemens and J. Morgan, On the mixed Hodge structure associated toπ3of a simply connected projective manifold, Ann. Sci. E.N.S.14(1981), 323-338. · Zbl 0511.14005
[11] D.-C. Cisinski and F. Déglise, Local and stable homological algebra in Grothendieck abelian categories, Homotopy Homology Applications,11(1), (2009), 219-260. · Zbl 1175.18007
[12] D.-C. Cisinski and F. Déglise, Mixed Weil cohomologies, Adv. Math.230, (2012), 55-130. · Zbl 1244.14014
[13] D.-C. Cisinski and F. Déglise, Triangulated Categories of Mixed Motives, Springer Monographs in Mathematics, 2019. · Zbl 07138952
[14] P. Deligne and A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ecole Norm. Sup., Serie 4 : Vol.38(2005), 1-56. · Zbl 1084.14024
[15] D.DuggerandD.C.Isaksen,Hypercoversintopology,preprintavailableat arXiv:math/0111287
[16] Y. Félix, S. Halperin, J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math. Springer, 2001.
[17] B. Fresse, Differential Graded Commutative Algebras and Cosimplicial Algebras, preprint version in (2014), Chapater II.6 of “Homotopy of operads and Grothendieck-Teichmullar groups”.
[18] H. Fukuyama and I. Iwanari, Monoidal infinity category of complexes from tannakian viewpoint, Math. Ann.356(2013), 519-553. · Zbl 1285.14017
[19] A. Grothendieck, Récoltes et Semailles, Gendai-Sugakusha, Japanese translation by Y. Tsuji, 1989.
[20] R. Hain, The de Rham Homotopy Theory of Complex Algebraic Variety I, K-theory1 (1987), 271-324. · Zbl 0637.55006
[21] K. Hess, Rational Homotopy Theory: A Brief Introduction,Interactions between homotopy theory and algebra, 175-202, Comtemp. Math., 436 Amer. Math. Soc., 2007. · Zbl 1128.55010
[22] V. Hinich, Homological algebra of homotopy algebras, Comm. in Algebra,25(1997), 3291- 3323. · Zbl 0894.18008
[23] M. Hovey, Model categories, Math. Survey and Monograph Vol. 83, 1999. · Zbl 0909.55001
[24] I. Iwanari, Tannakization in derived algebraic geometry, J. K-Theory,14(2014), 642-700. · Zbl 1325.18001
[25] I. Iwanari, Bar constructions and Tannakization, Publ. Res. Int. Math. Sci.,50(2014), 515-568. · Zbl 1308.18012
[26] I. Iwanari, Tannaka duality and stable infinity-categories, J. Topol.11(2018), 469-526. · Zbl 1423.14126
[27] I. Iwanari, On the structure of Galois groups of mixed motives, preprint, available at the author’s webpage https://sites.google.com/site/isamuiwanarishomepage/ · Zbl 1201.14002
[28] S. Kondo and S. Yasuda, Product structures in motivic cohomology and higher Chow groups, J. Pure Appl. Algebra215(2011), 511-522. · Zbl 1228.14020
[29] K. Künnemann, On the Chow motive of an abelian scheme, in:Motives(Seattle, WA 1991) 189-205, Proc. Symposia in Pure Math. Vol. 55 1994. · Zbl 0823.14032
[30] M. Levine, Tate motives and the vanishing conjectures for algebraic K-theory, (English summary) in:Algebraic K-theory and algebraic topology(Lake Louise, AB, 1991), 167-188, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993. · Zbl 0885.19001
[31] J.-L. Loday, Cyclic Homology, Springer-Verlag, 1998.
[32] J. Lurie, Higher topos theory, Ann. Math. Studies, 170 Princeton Univ. Press, 2009. · Zbl 1175.18001
[33] J. Lurie, Higher algebra, preprint, the version of September 2017, available at the author’s webpage.
[34] J. Lurie, Derived algebraic geometry series, preprint available at the author’s webpage.
[35] M. A. Mandell, Cochains and homotopy type, Publ. I.H.E.S.,103(2006), 213-246. · Zbl 1105.55003
[36] A. Mazel-Gee, Quillen adjucntions induce adjunction of quasicategories, preprint available atArXiv:1501.03146. · Zbl 1346.18003
[37] C. Mazza, V. Voevodsky and C. Weibel, Lecture Notes on Motivic cohomology, Clay Math. Monographs Vol. 2, 2006. · Zbl 1115.14010
[38] J. McClure and J. Smith, Multivariable cochain operations and littlen-cubes, J. Amer. Math. Soc.,16(2003), 681-704. · Zbl 1014.18005
[39] J. Morgan, The algebraic topology of smooth algebraic varieties, Publ. I.H.E.S.,48(1978), 137-204. · Zbl 0401.14003
[40] J.P. Murre, J. Nagel and C. Peters, Lectures on the Theory of Pure Motives, AMS University Lecture Series, Vol. 61, 2013. · Zbl 1273.14002
[41] M. Olsson, The bar construction and affine stacks, Comm. in Algebra,44(2016), 3088-3121. · Zbl 1348.18023
[42] F. Orgogozo, Isomotifs de dimension inférieure ou égale à un, manuscripta math.115, (2004), 339-360. · Zbl 1092.14027
[43] D. Quillen, Rational homotopy theory, Ann. Math.90(1969) 205-295 · Zbl 0191.53702
[44] O. Röndigs and P. A. Ostvaer, Modules over motivic cohomology, Adv. Math.219, (2008), 689-727. · Zbl 1180.14015
[45] A. J. Scholl, Classical motives, in:Motives (Seattle, WA 1991)163-187, Proc. Symposia in Pure Math. Vol. 55, 1994. · Zbl 0814.14001
[46] S. Schwede and B. Shipley, Equivalences of monoidal model categories, Algebraic and Geometric Topology Vol.3(2003), 287-334. · Zbl 1028.55013
[47] M. Spitzweck, Derived fundamental groups for Tate motives, preprint available at ArXiv:1005.2670
[48] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.E.S.,47(1977), 269- 331. · Zbl 0374.57002
[49] B. Toën, Champs affine, Selecta Math. (N.S.)12(2006), 39-135. · Zbl 1108.14004
[50] V. Voevodsky, Triangulated categories of motives over a field,Cycles, Transfers and motivichomology theories, 188-238, Ann. Math. Stud., 143 Princeton Univ. Press, 2000. · Zbl 1019.14009
[51] Z.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.