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

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S-1-action so that the quotient Y=X/S-1 is also a real algebraic variety. If pi:X --> Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with Q, pi* : (K) over bar (0)(R(Y, C)) circle times Q --> (K) over bar (0)(R(X, C)) circle times Q is onto, where R(X, C) = R(X) circle times C, R(X) denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak-Kucharz result that (K) over bar (0)(R(X x S-1, C)) = (K) over bar (0)(R(X, C)) for any real algebraic variety X. As an application we will show that for a compact connected Lie group G (K) over bar (0)(R(G, C)) circle times Q = 0. (C) 2002 Elsevier Science B.V. All rights reserved.