Algebraic & Geometric Topology

Coarse homology theories

Paul D Mitchener

Full-text: Open access


In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of “coarse CW–complexes”. This uniqueness result is used to prove a version of the coarse Baum–Connes conjecture for such spaces.

Article information

Algebr. Geom. Topol., Volume 1, Number 1 (2001), 271-297.

Received: 12 February 2001
Revised: 10 May 2001
Accepted: 16 May 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N35: Other homology theories 55N40: Axioms for homology theory and uniqueness theorems
Secondary: 19K56: Index theory [See also 58J20, 58J22] 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]

coarse geometry exotic homology coarse Baum–Connes conjecture Novikov conjecture


Mitchener, Paul D. Coarse homology theories. Algebr. Geom. Topol. 1 (2001), no. 1, 271--297. doi:10.2140/agt.2001.1.271.

Export citation


  • D.R. Anderson, F.X. Connolly, S.C. Ferry, E.K. Pedersen, Algebraic $K$-theory with continuous control at infinity, Journal of Pure and Applied Algebra 94 (1994) 25–47.
  • P. Baum, A. Connes, N. Higson, Classifying spaces for proper actions and $K$-theory of group $C^\star$-algebras, $C^\star$-algebras: 1943-1993 (S. Doran, ed.), Contemporary Mathematics, vol. 167, American Mathematical Society, 1994, pp. 241–291.
  • R.D. Christensen, H.J. Munkholm, Elements of controlled algebraic topology, In Preparation.
  • S. Eilenberg, N. Steenrod, Foundations of algebraic topology, Princeton University Press, 1952.
  • M. Gromov, Asymptotic invariants for infinite groups, Geometric Group Theory, Volume 2 (G.A. Niblo and M.A. Roller, eds.), London Mathematical Society Lecture Note Series, vol. 182, Cambridge University Press, 1993, pp. 1–295.
  • N. Higson, V. Lafforgue, G. Skandalis, Counterexamples to the Baum-Connes conjecture, Preprint (2001) available at:\nl
  • N. Higson, E.K. Pedersen, J. Roe, $C^\star$-algebras and controlled topology, $K$-theory 11 (1997) 209–239.
  • N. Higson, J. Roe, On the coarse Baum-Connes conjecture, Novikov Conjectures, Index Theorems, and Rigidity, Volume 2 (S.C.Ferry, A.Ranicki, and J.Rosenberg, eds.), London Mathematical Society Lecture Note Series, vol. 227, Cambridge University Press, 1995, pp. 227–254.
  • E.K. Pedersen, J. Roe, S. Weinberger, On the homotopy invariance of the boundedly controlled analytic signature over an open cone, Novikov Conjectures, Index Theorems, and Rigidity, Vol. 2 (S.C.Ferry, A.Ranicki, and J.Rosenberg, eds.), London Mathematical Society Lecture Note Series, vol. 227, Cambridge University Press, 1995, pp. 285–300.
  • J. Roe, Coarse cohomology and index theory on complete riemannian manifolds, Memoirs of the American Mathematical Society, vol. 497, American Mathematical Society, 1993.
  • J. Roe, Index theory, coarse geometry, and the topology of manifolds, Regional Conference Series on Mathematics, vol. 90, CBMS Conference Proceedings, American Mathematical Society, 1996.
  • G. Skandalis, J.L. Tu, G. Yu, Coarse Baum-Connes conjecture and groupoids, Preprint (2000), available at:\char'176tu/
  • E. Spanier, Algebraic topology, McGraw-Hill, 1966.
  • G. Yu, On the coarse Baum-Connes conjecture, $K$-theory 9 (1995) 199–221.
  • G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Inventiones Mathematicae 139 (2000).