Chern character for discrete groups.

*(English)*Zbl 0656.55005
A fête of topology, Pap. Dedic. Itiro Tamura, 163-232 (1988).

[For the entire collection see Zbl 0633.00020.]

The core of this significant expository paper is the definition of a Chern character for proper actions of a countable discrete group \(\Gamma\) on a \(C^{\infty}\)-manifold X. Let \(C_ 0(X)\) be the \(C^*\)-algebra of all continuous complex valued functions on X which vanish at \(\infty\), and \(C_ 0(X)\times \Gamma\) be the reduced cross-product algebra arising from the actionof \(\Gamma\) on \(C_ 0(X)\). Define \(K^ i_{\Gamma}(X)=K_ i[C_ 0(X)\times \Gamma]\) for \(i=0,1\). In order to get the correct definition for ordinary cohomology, let \(\hat X=\{(x,\gamma)\in X\times \Gamma:\) \(x\gamma =x\}\) with right \(\Gamma\)- action given by \((x,\gamma)\alpha =(x\alpha,\alpha^{-1}\gamma \alpha)\). By the finiteness of the isotropy subgroups the quotient space \^X/\(\Gamma\) has an orbifold structure - define \(H^ i(X,\Gamma)=\oplus_{j\in {\mathbb{N}}}H_ c^{2j+i}(\hat X/\Gamma;{\mathbb{C}})\) \((i=0,1)\). Using an isomorphism between these groups and the cyclic cohomology groups of the group algebra of \(\Gamma\) over \(C_ c^{\infty}(X)\), it is now possible to define an equivariant Chern character \(ch_{\Gamma}\), which induces an isomorphism between \(K^ i_{\Gamma}(X)\otimes {\mathbb{C}}\) and \(H^ i(X,\Gamma)\). (The two theories agree locally, both have Mayer-Vietoris sequences, and one can compare two spectral sequences of Segal type.) The relation with cyclic cohomology also leads to an index theorem for a \(\Gamma\)-equivariant family of elliptic operators. One approaches this central section of the paper through a motivational first part, in which the theory is developed for finite groups, only to realise that the authors are at heart interested in improper actions on X by a group which may be uncountable. They sidestep the problem of defining \(ch_{\Gamma}\) in this degree of generality by replacing \(K^ i_{\Gamma}(X)\) by groups \(K^ i(X,\Gamma)\) obtained by taking a suitable limit over manifolds W equivariantly submersed onto X, which admit proper actions. Instead of a single map \(ch_{\Gamma}\) one now obtains a commutative diagram \[ \begin{tikzcd} K_ i(X,\Gamma) \ar[r,"\mu"] \ar[d,"\varinjlim(ch_\Gamma)" '] & K_i[C_0(X)x\Gamma]\dar \\ H^i(X,\Gamma) \ar[r,"\bar\mu" '] & K_ i[C_0(X)x\Gamma] \otimes \mathbb{C}\rlap{\,,} \end{tikzcd} \] where the definition of \(\mu\) is forced on one by the limit groups \(K^ i\), and the right-hand vertical map is the obvious one. The authors conjecture that \(\mu\) is an isomorphism, and point out some of the profound implications this would have. These include the K-theoretic Novikov conjecture (since proved independently by I. Madsen and W. C. Hsiang), and in the case when \(\Gamma\) is torsion free, an isomorphism between \(K_ i(B\Gamma)\) and the corresponding algebraic K-group of the reduced \(C^*\)-algebra \(C^*_ r\Gamma\).

The core of this significant expository paper is the definition of a Chern character for proper actions of a countable discrete group \(\Gamma\) on a \(C^{\infty}\)-manifold X. Let \(C_ 0(X)\) be the \(C^*\)-algebra of all continuous complex valued functions on X which vanish at \(\infty\), and \(C_ 0(X)\times \Gamma\) be the reduced cross-product algebra arising from the actionof \(\Gamma\) on \(C_ 0(X)\). Define \(K^ i_{\Gamma}(X)=K_ i[C_ 0(X)\times \Gamma]\) for \(i=0,1\). In order to get the correct definition for ordinary cohomology, let \(\hat X=\{(x,\gamma)\in X\times \Gamma:\) \(x\gamma =x\}\) with right \(\Gamma\)- action given by \((x,\gamma)\alpha =(x\alpha,\alpha^{-1}\gamma \alpha)\). By the finiteness of the isotropy subgroups the quotient space \^X/\(\Gamma\) has an orbifold structure - define \(H^ i(X,\Gamma)=\oplus_{j\in {\mathbb{N}}}H_ c^{2j+i}(\hat X/\Gamma;{\mathbb{C}})\) \((i=0,1)\). Using an isomorphism between these groups and the cyclic cohomology groups of the group algebra of \(\Gamma\) over \(C_ c^{\infty}(X)\), it is now possible to define an equivariant Chern character \(ch_{\Gamma}\), which induces an isomorphism between \(K^ i_{\Gamma}(X)\otimes {\mathbb{C}}\) and \(H^ i(X,\Gamma)\). (The two theories agree locally, both have Mayer-Vietoris sequences, and one can compare two spectral sequences of Segal type.) The relation with cyclic cohomology also leads to an index theorem for a \(\Gamma\)-equivariant family of elliptic operators. One approaches this central section of the paper through a motivational first part, in which the theory is developed for finite groups, only to realise that the authors are at heart interested in improper actions on X by a group which may be uncountable. They sidestep the problem of defining \(ch_{\Gamma}\) in this degree of generality by replacing \(K^ i_{\Gamma}(X)\) by groups \(K^ i(X,\Gamma)\) obtained by taking a suitable limit over manifolds W equivariantly submersed onto X, which admit proper actions. Instead of a single map \(ch_{\Gamma}\) one now obtains a commutative diagram \[ \begin{tikzcd} K_ i(X,\Gamma) \ar[r,"\mu"] \ar[d,"\varinjlim(ch_\Gamma)" '] & K_i[C_0(X)x\Gamma]\dar \\ H^i(X,\Gamma) \ar[r,"\bar\mu" '] & K_ i[C_0(X)x\Gamma] \otimes \mathbb{C}\rlap{\,,} \end{tikzcd} \] where the definition of \(\mu\) is forced on one by the limit groups \(K^ i\), and the right-hand vertical map is the obvious one. The authors conjecture that \(\mu\) is an isomorphism, and point out some of the profound implications this would have. These include the K-theoretic Novikov conjecture (since proved independently by I. Madsen and W. C. Hsiang), and in the case when \(\Gamma\) is torsion free, an isomorphism between \(K_ i(B\Gamma)\) and the corresponding algebraic K-group of the reduced \(C^*\)-algebra \(C^*_ r\Gamma\).

Reviewer: Ch.Thomas

##### MSC:

55N91 | Equivariant homology and cohomology in algebraic topology |

57R20 | Characteristic classes and numbers in differential topology |

57S30 | Discontinuous groups of transformations |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

57S20 | Noncompact Lie groups of transformations |