Addendum to "Behnke-Stein Theorem for Analytic Spaces"

Abstract

A very simple argument shows that Theorem 3.1 in my paper Behnke-Stein theorem for analytic spaces, (these Transactions, 199 (1974), pp. 317 326) is enough, via a Narasimhan result, to obtain information about the torsion of the homology groups of a Runge pair of Stein spaces. Let (X, Y), with Y C X, Y open, be a pair of reduced complex analytic spaces of (complex) dimension n. Andreotti and Narasimhan proved in [1], among many others, the following result: (1.1) If (X, Y) is a Runge pair of Stein spaces (a 1-Runge pair in the terminology of [3] ) and every singularity of X outside Y is isolated, then Hr(X mod Y, Z) = 0 for r n + 1. We wish to show the following statements (1.2) and (1.3), (1.2) (Narasimhan [2, Theorem 3] ): if X is a Stein space, then: Hr(X, Z) = 0 for r > n + 1 and Hn(X, Z) is without torsion; (1.3) (Silva [3, Theorem 3.1]): if (X, Y) is a Runge pair of Stein spaces (or, equivalently, in the terminology of [3], a 1-Runge pair of cohomologically 1-complete spaces) then Hn+ (Xmod Y, C) =0; make us able to remove from (1.1) the assumption that the singularities of X outside Y are isolated. Indeed, if we write the exact homology sequence for the pair (X, Y): Received by the editors December 9, 1974. AMS (MOS) subject classifications (1970). Primary 32E 1 5.