On algebraic K-theory of real algebraic varieties with circle action

Abstract

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S1-action so that the quotient Y = X/S1 is also a real algebraic variety. If π:X → Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with ℚ, π*:K̃0(ℛ(Y,ℂ))⊗ℚ → K̃0(ℛ(X,ℂ))⊗ℚ is onto, where ℛ(X,ℂ)=ℛ(X)⊗ℂ, sr(X) denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak-Kucharz result that K̃0(ℛ(X × S1,ℂ))=K̃0(ℛ(X,ℂ)) for any real algebraic variety X. As an application we will show that for a compact connected Lie group G K̃0(ℛ(G,ℂ))⊗ℚ=0. © 2002 Elsevier Science B.V. All rights reserved.