# Equivariant geometric K-homology for compact Lie group actions

@article{Baum2010EquivariantGK,
title={Equivariant geometric K-homology for compact Lie group actions},
author={Paul Frank Baum and Herv'e Oyono-Oyono and Thomas Schick and Michael Walter},
journal={Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
year={2010},
volume={80},
pages={149-173}
}
• Published 3 February 2009
• Mathematics
• Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups $K^{G}_{*}(X)$, using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the “official” equivariant K-homology groups) and show that these are isomorphisms.
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## References

SHOWING 1-10 OF 35 REFERENCES
A Geometric Description of Equivariant K-Homology for Proper Actions
• Mathematics
• 2009
L etG be a discrete group and let X be a G-finite, proper G-CW-complex. We prove that Kasparov's equivariant K-homology groups KK G (C0(X),C) are isomor- phic to the geometric equivariant K-homology
The equivariant homotopy type of G-ANR’s for compact group actions
• Mathematics
• 2007
We prove that if G is a compact Hausdorff group then every G-ANR has the G-homotopy type of a G-CW complex. This is applied to extend the James–Segal G-homotopy equivalence theorem to the case of
Groups Acting on Buildings, Operator K-Theory, and Novikov's Conjecture
• Mathematics
• 1991
The paper is devoted to the study of the KK-theory of Bruhat-Tits buildings. We develop a theory which is analogous to the corresponding theory for manifolds of nonpositive sectional curvature. We
Equivariant K-theory
The purpose of this thesis is to present a fairly complete account of equivariant K-theory on compact spaces. Equivariant K-theory is a generalisation of K-theory, a rather well-known cohomology
TOPOLOGICAL INVARIANTS OF ELLIPTIC OPERATORS. I: K-HOMOLOGY
In this paper the homological K-functor is defined on the category of involutory Banach algebras, and Bott periodicity is proved, along with a series of theorems corresponding to the
On the Equivalence of Geometric and Analytic K-Homology
• Mathematics
• 2007
We give a proof that the geometric K-homology theory for finite CWcomplexes defined by Baum and Douglas is isomorphic to Kasparov’s Khomology. The proof is a simplification of more elaborate
A bordism-type description of homology
Abstract:Let h* be a generalized multiplicative cohomology theory. We give a description of the associated homology theory h* by means of h* and some kind of bordism. We describe the cap product and
THE OPERATOR K-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS
In this paper a general operator K-functor is constructed, depending on a pair A, B of C*-algebras. Special cases of this functor are the ordinary cohomological K-functor K*(B) and the homological
Equivariant K-theory
Topological K-theory [2] has many variants which have been developed and exploited for geometric purposes. There are real or quaternionic versions, “Real” K-theory in the sense of [1], equivariant