Koszul spaces
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Abstract:
We prove that a nilpotent space is both formal and coformal if and only if it is rationally homotopy equivalent to the derived spatial realization of a graded commutative Koszul algebra. We call such spaces Koszul spaces and show that the rational homotopy groups and the rational homology of iterated loop spaces of Koszul spaces can be computed by applying certain Koszul duality constructions to the cohomology algebra.References
- David J. Anick, A counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), no. 1, 1–33. MR 644015, DOI 10.2307/1971338
- David J. Anick, Comment: “A counterexample to a conjecture of Serre”, Ann. of Math. (2) 116 (1982), no. 3, 661. MR 678485, DOI 10.2307/2007027
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- H. J. Baues and J.-M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), no. 3, 219–242. MR 431172, DOI 10.1007/BF01425239
- J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609
- A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
- Victor M. Buchstaber and Taras E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, vol. 24, American Mathematical Society, Providence, RI, 2002. MR 1897064, DOI 10.1090/ulect/024
- F. R. Cohen, The homology of $\mathcal {C}_{n+1}$-spaces, $n\geq 0$, in “The homology of iterated loop spaces”, Lecture Notes in Mathematics 533, Springer Verlag (1976), 207–351.
- F. R. Cohen and S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748. MR 1881013, DOI 10.1090/S0002-9947-02-02948-3
- Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. MR 1104531, DOI 10.1215/S0012-7094-91-06217-4
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
- Yves Félix, John Oprea, and Daniel Tanré, Algebraic models in geometry, Oxford Graduate Texts in Mathematics, vol. 17, Oxford University Press, Oxford, 2008. MR 2403898
- Benoit Fresse, Koszul duality of operads and homology of partition posets, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 115–215. MR 2066499, DOI 10.1090/conm/346/06287
- Ralph Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39. MR 404254, DOI 10.7146/math.scand.a-11585
- E. Getzler, J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv:hep-th/9403055v1
- Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MR 1301191, DOI 10.1215/S0012-7094-94-07608-4
- P. Lambrechts, I. Volic, Formality of the little $N$-disks operad, arXiv:0808.0457v2 [math.AT]
- Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392, DOI 10.1007/978-3-642-30362-3
- Martin Majewski, Rational homotopical models and uniqueness, Mem. Amer. Math. Soc. 143 (2000), no. 682, xviii+149. MR 1751423, DOI 10.1090/memo/0682
- J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 0420610
- Joan Millès, The Koszul complex is the cotangent complex, Int. Math. Res. Not. IMRN 3 (2012), 607–650. MR 2885984, DOI 10.1093/imrn/rnr034
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
- Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. 74 (1978), no. 2, 429–460. MR 494641
- Joseph Neisendorfer, The rational homotopy groups of complete intersections, Illinois J. Math. 23 (1979), no. 2, 175–182. MR 528555
- Joseph Neisendorfer and Timothy Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978), no. 4, 565–580. MR 500938
- Dietrich Notbohm and Nigel Ray, On Davis-Januszkiewicz homotopy types. I. Formality and rationalisation, Algebr. Geom. Topol. 5 (2005), 31–51. MR 2135544, DOI 10.2140/agt.2005.5.31
- Taras E. Panov and Nigel Ray, Categorical aspects of toric topology, Toric topology, Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, 2008, pp. 293–322. MR 2428364, DOI 10.1090/conm/460/09026
- Stefan Papadima and Alexander I. Suciu, Homotopy Lie algebras, lower central series and the Koszul property, Geom. Topol. 8 (2004), 1079–1125. MR 2087079, DOI 10.2140/gt.2004.8.1079
- Stefan Papadima and Sergey Yuzvinsky, On rational $K[\pi ,1]$ spaces and Koszul algebras, J. Pure Appl. Algebra 144 (1999), no. 2, 157–167. MR 1731434, DOI 10.1016/S0022-4049(98)00058-9
- Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MR 265437, DOI 10.1090/S0002-9947-1970-0265437-8
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
- Jean-Pierre Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. of Math. (2) 58 (1953), 258–294 (French). MR 59548, DOI 10.2307/1969789
- Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468
- Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078
- B. Vallette, Homotopy theory of homotopy algebras, preprint.
Additional Information
- Alexander Berglund
- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitets- parken 5, 2100 Copenhagen Ø, Denmark
- Address at time of publication: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
- Email: alexb@math.ku.dk
- Received by editor(s): November 18, 2011
- Received by editor(s) in revised form: August 8, 2012
- Published electronically: April 16, 2014
- Additional Notes: This work was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 4551-4569
- MSC (2010): Primary 55P62; Secondary 16S37
- DOI: https://doi.org/10.1090/S0002-9947-2014-05935-7
- MathSciNet review: 3217692