Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T09:12:58.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomology rings of almost-direct products of free groups

Part of: Classical Topics in Algebraic Topology Connections with homological algebra and category theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  22 January 2010

Daniel C. Cohen
Daniel C. Cohen*
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA (email: cohen@math.lsu.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg–MacLane space.

Keywords

almost-direct productcohomology ringtopological complexity

MSC classification

Secondary: 20J06: Cohomology of groups 20F28: Automorphism groups of groups 20F36: Braid groups; Artin groups 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 465 - 479
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Aramova, A., Herzog, J. and Hibi, T., Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), 174–211; MR 1444495.CrossRefGoogle Scholar
[2]Arnol’d, V. I., The cohomology ring of the group of dyed braids, Math. Notes 5 (1969), 138–140; MR 0242196.CrossRefGoogle Scholar
[3]Birman, J., Braids, links and mapping class groups, Annals of Mathematical Studies, vol. 82 (Princeton University Press, Princeton, NJ, 1975); MR 375281.CrossRefGoogle Scholar
[4]Cohen, F. R., The homology of 𝒞n+1-spaces, n≥0, in The homology of iterated loop spaces, Lecture Notes in Mathematics, vol. 533 (Springer, Berlin, 1976), 207–352; MR 0436146.CrossRefGoogle Scholar
[5]Cohen, D. C., Topological complexity of almost-direct products of free groups, Oberwolfach Rep. 4 (2007), 2331–2334; MR 2432117.Google Scholar
[6]Cohen, F. R., Pakianathan, J., Vershinin, V. and Wu, J., Basis-conjugating automorphisms of a free group and associated Lie algebras, in Groups, homotopy and configuration spaces (Tokyo 2005), Geometry and Topology Monographs, vol. 13 (Geom. Topol. Publ., Coventry, 2008), 147–168; MR 2508204.Google Scholar
[7]Cohen, D. C. and Pruidze, G., Topological complexity of basis-conjugating automorphism groups, Pacific J. Math. 238 (2008), 233–248; MR 2442993.CrossRefGoogle Scholar
[8]Cohen, D. C. and Suciu, A., Homology of iterated semidirect products of free groups, J. Pure Appl. Algebra 126 (1998), 87–120; MR 1600518.CrossRefGoogle Scholar
[9]Eilenberg, S. and Ganea, T., On the Lusternik–Schnirelmann category of abstract groups, Ann. of Math. (2) 65 (1957), 517–518; MR 0085510.CrossRefGoogle Scholar
[10]Falk, M. and Randell, R., The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77–88; MR 0808110.Google Scholar
[11]Falk, M. and Randell, R., Pure braid groups and products of free groups, in Braids, Contemporary Mathematics, vol. 78 (American Mathematical Society, Providence, RI, 1988), 217–228; MR 0975081.CrossRefGoogle Scholar
[12]Farber, M., Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221; MR 1957228.CrossRefGoogle Scholar
[13]Farber, M., Instabilities of robot motion, Topology Appl. 140 (2004), 245–266; MR 2074919.CrossRefGoogle Scholar
[14]Farber, M., Topology of robot motion planning, in Morse theoretic methods in non-linear analysis and in symplectic topology, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 217 (Springer, Berlin, 2006), 185–230; MR 2276952.Google Scholar
[15]Farber, M., Grant, M. and Yuzvinsky, S., Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles, in Topology and robotics, Contemporary Mathematics, vol. 438 (American Mathematical Society, Providence, RI, 2007), 75–83; MR 2359030.CrossRefGoogle Scholar
[16]Farber, M. and Yuzvinsky, S., Topological robotics: subspace arrangements and collision free motion planning, in Geometry, topology, and mathematical physics, American Mathematical Society Translations, Series 2, vol. 212 (American Mathematical Society, Providence, RI, 2004), 145–156; MR 2070052.Google Scholar
[17]James, I. M., On category, in the sense of Lusternik–Schnirelmann, Topology 17 (1978), 331–348; MR 0516214.CrossRefGoogle Scholar
[18]Kohno, T., Série de Poincaré–Koszul associée aux groupes de tresses pures, Invent. Math. 179 (1985), 57–75; MR 0808109.Google Scholar
[19]Latombe, J.-C., Robot motion planning (Kluwer, Dordrecht, 1991).Google Scholar
[20]Magnus, W., Karras, A. and Solitar, D., Combinatorial group theory: representations of groups in terms of generators and relations, Pure and Applied Mathematics, vol. 13 (Interscience Publishers, New York, 1966); MR 0207802.Google Scholar
[21]McCleary, J., A user’s guide to spectral sequences, second edition, Cambridge Studies in Advanced Mathematics, vol. 58 (Cambridge University Press, Cambridge, 2001); MR 1793722.Google Scholar
[22]Orlik, P. and Terao, H., Arrangements of hyperplanes, Grundlehrender Mathematischen Wissenschaften, vol. 300 (Springer, Berlin, 1992); MR 1217488.CrossRefGoogle Scholar
[23]Paris, L., On the fundamental group of the complement of a complex hyperplane arrangement, in Arrangements – Tokyo 1998, Advanced Studies in Pure Mathematics, vol. 27 (Kinokuniya, Tokyo, 2000), 257–272; MR 1796904.Google Scholar
[24]Sharir, M., Algorithmic motion planning, in Handbook of discrete and computational geometry, CRC Press Series: Discrete Mathematics Applications (CRC Press, Boca Raton, FL, 1997), 733–754; MR 1730196.Google Scholar
[25]Shelton, B. and Yuzvinsky, S., Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997), 477–490; MR 1610447.Google Scholar
[26]Yuzvinsky, S., Orlik–Solomon algebras in algebra and topology, Russian Math. Surveys 56 (2001), 293–364; MR 1859708.CrossRefGoogle Scholar