# zbMATH — the first resource for mathematics

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 $$C^*_ r\Gamma$$.
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