nLab
Baum-Connes conjecture

Contents

Contents

Idea

The Baum-Connes conjecture asserts that under suitable technical conditions, the operator K-theory/KK-theory of the groupoid convolution algebra of an action topological groupoid X//GX//G is equivalent to the GG-equivariant topological K-theory/equivariant operator K-theory/equivariant KK-theory of XX. Moreover, it says that this equivalences is exhibited by the analytic assembly map. (Just the injectivity of this map is related to the Novikov conjecture.)

The original version of the Baum-Connes conjecture (Baum-Connes) stated for a suitable topological group GG that with X=EG*X = E G \simeq \ast the point incarnated as the GG-universal principal bundle with its free GG-action the analytic assembly map (the GG-equivariant index map)

K G(EG)K (C(BG)), K_\bullet^G(E G) \stackrel{}{\to} K_\bullet(C(\mathbf{B}G)) \,,

from the GG-equivariant K-theory of EGE G to the operator K-theory of the group algebra, hence the groupoid convolution algebra of the delooping groupoid BG*//G\mathbf{B}G \simeq \ast // G, is an isomorphism.

This Baum-Connes conjecture is known to be true for

It is not known if the conjecture is true for all discrete groups.

Later the statement was generalized (Tu 99) to more general groupoids.

In (Kasparov 88) the refinement of the analytic assembly map to equivariant KK-theory is given, and called the descent map. This is of the form (recalled as Blackadar, theorem 20.6.2)

KK G(A,B)KK(GA,GB) KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)

where on the left we have GG-equivariant KK-theory and on the right ordinary KK-theory of crossed product C*-algebras (which by the discussion there are models for the groupoid convolution algebras of GG-action groupoids).

This is an isomorphism at least for GG a compact topological group and restricted to operator K-theory (hence to the first argument being \mathbb{C}) and for GG a discrete group and restricted to K-homology (hence ot the second argument being \mathbb{C}). In this form this is the Green-Julg theorem, see below.

Theorems

Green-Rosenberg-Julg theorem

The Green-Rosenberg-Julg theorem identifies equivariant K-theory with the operator K-theory of crossed product algebras.

Theorem

(Green-Julg theorem)

Let GG be a topological group acting on a C*-algebra AA.

  1. If GG is a compact topological group then the descent map

    KK G(,A)KK(,GA) KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)

    is an isomorphism, identifying the equivariant operator K-theory of AA with the ordinary operator K-theory of the crossed product C*-algebra GAG \ltimes A.

  2. if GG is a discrete group then the descent map

    KK G(A,)KK(GA,) KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})

    is an isomorphism, identifying the equivariant K-homology of AA with the ordinary K-homology of the crossed product C*-algebra GAG \ltimes A.

This goes back to (Green 82), (Julg 81). A KK-theory-proof is in (Echterhoff, theorem 0.2); a textbook account is in (Blackadar, 11.7, 20.2.7). See also around (Land 13, prop. 41).

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
(equivariant)
stable cohomotopy
K 𝔽 1Segal 74\mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)

References

Introductions and surveys

Introductions and surveys include

Textbook discussion is in sections 11.7 and 20.2.7 of

See also

Original articles

The original article is

Proof of the conjecture for hyperbolic groups is in

  • Vincent Lafforgue, La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques, Journal of Noncommutative Geometry, Volume 6, Issue 1, 2012, pages 1-197 (arXiv:1201.4653)

Technical subtleties are discussed in

The generalization to Lie groupoids is due to

Proofs for some cases are in

KK-theory tools and the descent map are introduced in

  • Gennady Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Mathematicae, vol. 91, p.147, 1988(web)

The “Green-Julg theorem” for commutative algebra and finite group is due to Michael Atiyah, for commutative algebra and general group due to

  • P. Green, Equivariant if-theory and crossed product C- algebras_, pp. 337-338 in Operator algebras and applications (Kingston, Ont., 1980), vol. 1, edited by R. V. Kadison, Proc. Sympos. Pure Math. 38, Amer. Math. Soc, Providence, 1982.

and the general case is due to an unpublished result by Green and Jonathan Rosenberg and independently due to

  • P. Julg, K-theorie equivariante et produits croises, C. R. Acad. Sci. Paris Ser. I Math. 292:13 1981, 629-632.

Further discussion is in

  • Siegfried Echterhoff, The Green-Julg theorem pdf
  • Walther Paravicini, A generalised Green-Julg theorem for proper groupoids and Banach algebras, (arXiv:0902.4365)

  • V. Lafforgue, pdf

A modification of the Baum-Connes conjecture with coefficient where many counterexamples (to the conjecture with coefficients) are eliminated is in

  • Paul Baum, Erik Guentner, Rufus Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, arxiv/1311.2343

Discussion in terms of localization/homotopy theory is in

  • Ralf Meyer, Ryszard Nest, The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259.

Last revised on September 10, 2018 at 08:37:15. See the history of this page for a list of all contributions to it.