Signature Map and Unramified Cohomology
by
Jean-Philippe Monnier
University of Angers

Let X be a variety of dimension d over a real closed field R. The total signature homomorphism \Lambda: W(X) --> Cont(X(R), Z) is defined on the Witt ring of X with values in the set of continuous functions from the real points of X to Z. Parimala and Mahé asked the question whether the exponent of cokernel of \Lambda is bounded by 2^d+1, for smooth varieties of dimension d over R. We answer positively to this question when d <= 2 and d <= 3 in the affine case. Moreover, for smooth surfaces we prove that 4.Cont(X(R), Z) subset or equal Im \Lambda if and only if the graded Witt ring is isomorphic to the graded unramified cohomology ring. For real rational surfaces, we show that the previous condition is fulfilled and we describe completely the image of \Lambda.

Date received: February 10, 1999