## Geometry & Topology

### Homotopy Lie algebras, lower central series and the Koszul property

#### Abstract

Let $X$ and $Y$ be finite-type CW–complexes ($X$ connected, $Y$ simply connected), such that the rational cohomology ring of $Y$ is a $k$–rescaling of the rational cohomology ring of $X$. Assume $H∗(X,ℚ)$ is a Koszul algebra. Then, the homotopy Lie algebra $π∗(ΩY)⊗ℚ$ equals, up to $k$–rescaling, the graded rational Lie algebra associated to the lower central series of $π1(X)$. If $Y$ is a formal space, this equality is actually equivalent to the Koszulness of $H∗(X,ℚ)$. If $X$ is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of $π1(X)$ and the completion of $[ΩS2k+1,ΩY]$. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

#### Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1079-1125.

Dates
Accepted: 17 July 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883463

Digital Object Identifier
doi:10.2140/gt.2004.8.1079

Mathematical Reviews number (MathSciNet)
MR2087079

Zentralblatt MATH identifier
1127.55004

#### Citation

Papadima, Ştefan; Suciu, Alexander I. Homotopy Lie algebras, lower central series and the Koszul property. Geom. Topol. 8 (2004), no. 3, 1079--1125. doi:10.2140/gt.2004.8.1079. https://projecteuclid.org/euclid.gt/1513883463

#### References

• D Anick, Connections between Yoneda and Pontrjagin algebras, from: “Algebraic topology (Aarhus, 1982)” Lecture Notes in Math. 1051, Springer, Berlin (1984) 331–350
• H J Baues, Commutator calculus and groups of homotopy classes, London Math. Soc. Lecture Note Series 50, Cambridge Univ. Press, Cambridge–New York (1981)
• A Beilinson, V Ginzburg, W Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996) 473–527
• A K Bousfield, V K A M Gugenheim, On PL De Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 no 179 (1976)
• B Berceanu, S Papadima, Cohomologically generic $2$–complexes and $3$–dimensional Poincaré complexes, Math. Ann. 298 (1994) 457–480
• R Bott, H Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1953) 320–337
• E Brieskorn, Sur les groupes de tresses, Séminaire Bourbaki, 1971/72, Lecture Notes in Math. 317, Springer, Berlin (1973) 21–44
• D C Cohen, F R Cohen, M Xicoténcatl, Lie algebras associated to fiber-type arrangements, Intern. Math. Res. Not. 2003:29 (2003) 1591–1621
• F R Cohen, S Gitler, Loop spaces of configuration spaces, braid-like groups, and knots, from: “Cohomological methods in homotopy theory (Bellaterra, 1998)” Progress in Math. vol. 196, Birkhäuser, Basel (2001) 59–78
• F R Cohen, S Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002) 1705–1748
• F R Cohen, T Kohno, M Xicoténcatl, Orbit configuration spaces associated to discrete subgroups of $\operatorname{PSL}(2, \R)$.
• F R Cohen, T Sato, On groups of homotopy groups, loop spaces, and braid-like groups, preprint (2001)
• F R Cohen, M Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: homology and homotopy groups, Topology Appl. 118 (2002) 17–29
• C De Concini, C Procesi, Wonderful models of subspace arrangements, Selecta Math. 1 (1995) 459–494
• P Deligne, P Griffiths, J Morgan, D Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975) 245–274
• E Fadell, S Husseini, Geometry and topology of configuration spaces, Springer Monographs in Math. Springer–Verlag, Berlin (2001)
• M Falk, R Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985) 77–88
• A Hattori, Topology of $\C^n$ minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975) 205–219
• P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies 15, North-Holland, Amsterdam-Oxford; Elsevier, New York (1975)
• M Jambu, S Papadima, A generalization of fiber-type arrangements and a new deformation method, Topology 37 (1998) 1135–1164
• M Kervaire, An interpretation of G Whitehead's generalization of H Hopf's invariant, Ann. of Math. 69 (1959) 345–365
• T Kohno, Série de Poincaré–Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985) 57–75
• U Koschorke, D Rolfsen, Higher dimensional link operations and stable homotopy, Pacific J. Math. 139 (1989) 87–106
• J Labute, On the descending central series of groups with a single defining relation, J. Algebra 14 (1970) 16–23
• M Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup. 71 (1954) 101–190
• M Markl, S Papadima, Homotopy Lie algebras and fundamental groups via deformation theory, Annales Inst. Fourier (Grenoble) 42 (1992) 905–935
• J W Milnor, Morse theory, Ann. Math. Studies 51, Princeton Univ. Press, Princeton, NJ (1963)
• J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. Math. 81 (1965) 211–264
• J Neisendorfer, T Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978) 565–580
• A Némethi, The link of the sum of two holomorphic functions, Proceedings of the National Conference on Geometry and Topology (Tîrgovişte, 1986) Univ. Bucureşti, Bucharest (1988) 181–184
• M Oka, On the fundamental group of the complement of certain plane curves, J. Math. Soc. Japan 30 (1978) 579–597
• S Papadima, The cellular structure of formal homotopy types, J. Pure Appl. Algebra 35 (1985) 171–184
• S Papadima, Campbell–Hausdorff invariants of links, Proc. London Math. Soc. 75 (1997) 641–670
• S Papadima, Braid commutators and homogeneous Campbell–Hausdorff tests, Pacific J. Math. 197 (2001) 383–416
• S Papadima, A Suciu, Rational homotopy groups and Koszul algebras, C. R. Math. Acad. Sci. Paris 335 (2002) 53–58
• S Papadima, A Suciu, Chen Lie algebras, Int. Math. Res. Not. 2004:21 (2004) 1057–1086
• S Papadima, S Yuzvinsky, On rational $K[\pi,1]$ spaces and Koszul algebras, J. Pure Appl. Algebra 144 (1999) 157–167
• G Porter, Higher order Whitehead products, Topology 3 (1965) 123–135
• S Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970) 39–60
• D Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205–295
• D Rolfsen, Knots and links, Math. Lecture Series 7, Publish or Perish, Berkeley, CA (1976)
• J-E Roos, On the characterization of Koszul algebras. Four counter-examples, C. R. Acad. Sci. Paris, Série I, 321 (1995) 15–20
• T Sato, On the group of morphisms of coalgebras, Ph.D. thesis, University of Rochester, New York (2000)
• H Scheerer, On rationalized $H$– and co-$H$–spaces. With an appendix on decomposable $H$– and co-$H$–spaces, Manuscripta Math. 51 (1985) 63–87
• H Schenck, A Suciu, Lower central series and free resolutions of hyperplane arrangements, Trans. Amer. Math. Soc. 354 (2002) 3409–3433
• J-P Serre, Lie groups and Lie algebras, Second edition, Lecture Notes in Math. 1500, Springer–Verlag, Berlin (1992)
• B Shelton, S Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997) 477–490
• H Shiga, N Yagita, Graded algebras having a unique rational homotopy type, Proc. Amer. Math. Soc. 85 (1982) 623–632
• T Stanford, Braid commutators and Vassiliev invariants, Pacific J. Math. 174 (1996) 269–276
• D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 269–331
• D Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lecture Notes in Math. 1025, Springer–Verlag, Berlin (1983)
• G W Whitehead, Elements of homotopy theory, Grad. Texts in Math. 61, Springer–Verlag, New York, Berlin (1978)
• S Yuzvinsky, Orlik–Solomon algebras in algebra and topology, Russian Math. Surveys 56 (2001) 293–364
• S Yuzvinsky, Small rational model of subspace complement, Trans. Amer. Math. Soc. 354 (2002) 1921–1945