Algebraic & Geometric Topology

Hochschild homology relative to a family of groups

Andrew Nicas and David Rosenthal

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We define the Hochschild homology groups of a group ring G relative to a family of subgroups of G. These groups are the homology groups of a space which can be described as a homotopy colimit, or as a configuration space, or, in the case is the family of finite subgroups of G, as a space constructed from stratum preserving paths. An explicit calculation is made in the case G is the infinite dihedral group.

Article information

Algebr. Geom. Topol., Volume 8, Number 2 (2008), 693-728.

Received: 19 September 2007
Revised: 31 January 2008
Accepted: 12 February 2008
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 55R35: Classifying spaces of groups and $H$-spaces 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]

Hochschild homology family of subgroups classifying space


Nicas, Andrew; Rosenthal, David. Hochschild homology relative to a family of groups. Algebr. Geom. Topol. 8 (2008), no. 2, 693--728. doi:10.2140/agt.2008.8.693.

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  • R Ayala, F F Lasheras, A Quintero, The equivariant category of proper $G$–spaces, Rocky Mountain J. Math. 31 (2001) 1111–1132
  • H Biller, Characterizations of proper actions, Math. Proc. Cambridge Philos. Soc. 136 (2004) 429–439
  • C J R Borges, On stratifiable spaces, Pacific J. Math. 17 (1966) 1–16
  • N Bourbaki, Elements of mathematics. General topology. Part 1, Hermann, Paris (1966)
  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Math. 46, Academic Press, New York (1972)
  • R Cauty, Sur les espaces d'applications dans les CW–complexes, Arch. Math. $($Basel$)$ 27 (1976) 306–311
  • J F Davis, W Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$– and $L$–theory, $K$–Theory 15 (1998) 201–252
  • J J Duistermaat, J A C Kolk, Lie groups, Universitext, Springer, Berlin (2000)
  • R Geoghegan, A Nicas, Parametrized Lefschetz–Nielsen fixed point theory and Hochschild homology traces, Amer. J. Math. 116 (1994) 397–446
  • A Hatcher, F Quinn, Bordism invariants of intersections of submanifolds, Trans. Amer. Math. Soc. 200 (1974) 327–344
  • B Hughes, Stratified path spaces and fibrations, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 351–384
  • J-L Loday, Cyclic homology, second edition, Grundlehren series 301, Springer, Berlin (1998)
  • W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Math. 1408, Springer, Berlin (1989) Mathematica Gottingensis
  • W Lück, Survey on classifying spaces for families of subgroups, from: “Infinite groups: geometric, combinatorial and dynamical aspects”, Progr. Math. 248, Birkhäuser, Basel (2005) 269–322
  • W Lück, H Reich, Detecting $K$–theory by cyclic homology, Proc. London Math. Soc. $(3)$ 93 (2006) 593–634
  • A Lundell, S Weingram, The topology of CW complexes, Van Nostrand Reinhold Co., New York (1969)
  • H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. $(2)$ 120 (1984) 39–87
  • J Milnor, Construction of universal bundles. I, Ann. of Math. $(2)$ 63 (1956) 272–284
  • R S Palais, The classification of $G$–spaces, Mem. Amer. Math. Soc. 36 (1960) iv+72
  • R S Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. $(2)$ 73 (1961) 295–323
  • E H Spanier, Algebraic topology, McGraw-Hill Book Co., New York (1966)
  • R N Talbert, An isomorphism between Bredon and Quinn homology, Forum Math. 11 (1999) 591–616