Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Koszul spaces


Author: Alexander Berglund
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
Published electronically: April 16, 2014
MathSciNet review: 3217692
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] David J. Anick, A counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), no. 1, 1-33. MR 644015 (86i:55011a), https://doi.org/10.2307/1971338
  • [2] David J. Anick, Comment: ``A counterexample to a conjecture of Serre'', Ann. of Math. (2) 116 (1982), no. 3, 661. MR 678485 (86i:55011b), https://doi.org/10.2307/2007027
  • [3] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [4] H. J. Baues and J.-M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), no. 3, 219-242. MR 0431172 (55 #4174)
  • [5] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin, 1973. MR 0420609 (54 #8623a)
  • [6] A. K. Bousfield and V. K. A. M. Gugenheim, On $ {\rm PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 0425956 (54 #13906)
  • [7] 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 (2003e:57039)
  • [8] 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.
  • [9] F. R. Cohen and S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705-1748. MR 1881013 (2002m:55020), https://doi.org/10.1090/S0002-9947-02-02948-3
  • [10] Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417-451. MR 1104531 (92i:52012), https://doi.org/10.1215/S0012-7094-91-06217-4
  • [11] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 (2002d:55014)
  • [12] 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 (2009a:55006)
  • [13] 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 (2005g:18015), https://doi.org/10.1090/conm/346/06287
  • [14] Ralph Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29-39. MR 0404254 (53 #8057)
  • [15] E. Getzler, J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv:hep-th/9403055v1
  • [16] Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203-272. MR 1301191 (96a:18004), https://doi.org/10.1215/S0012-7094-94-07608-4
  • [17] P. Lambrechts, I. Volic, Formality of the little $ N$-disks operad, arXiv:0808.0457v2 [math.AT]
  • [18] Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392
  • [19] Martin Majewski, Rational homotopical models and uniqueness, Mem. Amer. Math. Soc. 143 (2000), no. 682, xviii+149. MR 1751423 (2001e:55015)
  • [20] J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin, 1972. Lectures Notes in Mathematics, Vol. 271. MR 0420610 (54 #8623b)
  • [21] Joan Millès, The Koszul complex is the cotangent complex, Int. Math. Res. Not. IMRN 3 (2012), 607-650. MR 2885984
  • [22] John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 0174052 (30 #4259)
  • [23] Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. 74 (1978), no. 2, 429-460. MR 494641 (80b:55010)
  • [24] Joseph Neisendorfer, The rational homotopy groups of complete intersections, Illinois J. Math. 23 (1979), no. 2, 175-182. MR 528555 (80j:55018)
  • [25] Joseph Neisendorfer and Timothy Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978), no. 4, 565-580. MR 0500938 (58 #18429)
  • [26] Dietrich Notbohm and Nigel Ray, On Davis-Januszkiewicz homotopy types. I. Formality and rationalisation, Algebr. Geom. Topol. 5 (2005), 31-51 (electronic). MR 2135544 (2006a:55016), https://doi.org/10.2140/agt.2005.5.31
  • [27] 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 (2009j:55026), https://doi.org/10.1090/conm/460/09026
  • [28] Stefan Papadima and Alexander I. Suciu, Homotopy Lie algebras, lower central series and the Koszul property, Geom. Topol. 8 (2004), 1079-1125. MR 2087079 (2005g:55022), https://doi.org/10.2140/gt.2004.8.1079
  • [29] 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 (2000k:55017), https://doi.org/10.1016/S0022-4049(98)00058-9
  • [30] Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39-60. MR 0265437 (42 #346)
  • [31] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295. MR 0258031 (41 #2678)
  • [32] Jean-Pierre Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math. (2) 58 (1953), 258-294 (French). MR 0059548 (15,548c)
  • [33] Jean-Pierre Serre, Algèbre locale. Multiplicités, Cours au Collège de France, 1957-1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin, 1965 (French). MR 0201468 (34 #1352)
  • [34] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331 (1978). MR 0646078 (58 #31119)
  • [35] B. Vallette, Homotopy theory of homotopy algebras, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 55P62, 16S37

Retrieve articles in all journals with MSC (2010): 55P62, 16S37


Additional Information

Alexander Berglund
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 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

DOI: https://doi.org/10.1090/S0002-9947-2014-05935-7
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society