93i:55018b 55R35 19L47 55R50

Adem, Alejandro ^{(}1-WI^{)}

Erratum: “On the K-theory of the classifying space of a discrete group”. Math. Ann. 293 (1992), no. 2, 385–386. From the introduction: “Let G be a finite group. By considering the action of G on itself by conjugation, one arrives at the class equation for G, (1) 1 − 1/|G| = ∑_{(g),g≠1} 1/|C(g)|, where g ranges over conjugacy classes of elements in G, with centralizer C(g).

“Now assume that Γ is a discrete group of finite virtual cohomological dimension with centralizers of elements of finite order in Γ homologically finite (for example SL_{n}(Z)). K. S. Brown [J. Pure Appl. Algebra 24 (1982), no. 2, 103–121; MR 83d:20035] proved the generalization of (0.1) to Γ: (2) χ̃(Γ) − χ(Γ) = ∑_{(γ),γ≠1} χ(C(γ)), where γ ranges over conjugacy classes of elements of finite order in Γ, χ̃ denotes the topological Euler characteristic and χ the usual grouptheoretic version (if Γ′ is a torsion-free subgroup of finite index in Γ then by definition χ(Γ) = χ̃(Γ′)/[Γ : Γ′]). The point of (0.2) is that the difference between χ̃(Γ) and χ(Γ) is determined by the torsion in Γ; in particular, if Γ is torsion-free they coincide.

“Now let us recall a well-known result in algebra: Let R(G) denote the character ring of a finite group G. Then (3) dim_{C} R(G) ⊗ C − 1 = the number of conjugacy classes of nontrivial elements in G. The term on the left can be identified with dim_{C} K_{G}^{∗}(∗) ⊗ C, the dimension of the equivariant K-theory of a point. In this paper we use this interpretation to outline an approach for generalizing this formula to infinite groups Γ of finite virtual cohomological dimension (with suitable finiteness conditions), motivated by (2).

“By a result due to J.-P. Serre [in Prospects in mathematics (Princeton, NJ, 1970), 77–169, Ann. of Math. Stud., 70, Princeton Univ. Press, Princeton, NJ, 1971; MR 525876], Γ will act on a finitedimensional, contractible space X, with finite isotropy, and such that X^{H} ≃ ∗ for all H ⊆ Γ finite. Fix an extension 1 → Γ′ → Γ →^{π} G → 1 with Γ′ torsion-free, G finite. Then Γ′ acts freely on X and G acts on X/Γ′ such that the finite subgroups in Γ correspond under π to the isotropy subgroups in G.

“We have the following: Theorem 3.1.

K_{G}^{∗}(X/Γ′) ⊗ C ≅ K^{∗}(BΓ)^{G} ⊗ C ⊕ ⨁_{γ∈𝒞−{1}} K^{∗}(B(C(γ) ∩ Γ′))^{H_{γ}} ⊗ C,

where 𝒞 is the set of conjugacy classes of elements of finite order in Γ and H_{γ} = C(γ)/C(γ) ∩ Γ′, a finite group. An immediate corollary of this is the formula (4) χ(K_{G}^{∗}(X/Γ′) ⊗ C) − χ̃(Γ) = ∑_{γ∈𝒞−{1}} χ̃(C(γ)), where χ(K_{G}^{∗}(X/Γ′) ⊗ C) = dim_{C} K_{G}^{0}(X/Γ′) ⊗ C − dim_{C} K_{G}^{1}(X/Γ′) ⊗ C.

“The paper is organized as follows. In Section 1 we describe the Γcomplex X; in Section 2 we outline the main properties of equivariant K-theory; in Section 3 we prove our results and in Section 4 we provide examples, as well as a p-local version of the results in Section 3; using p-adic K-theory we obtain an exact formula (4.2).”

In the erratum the author corrects the statement of Theorem 3.1 (the above summary reflects the change).

\begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 1993, 1996