The purpose of this paper is to compare the algebraic Witt group $$W(V)$$ of quadratic forms for an algebraic variety $$V$$ over $$ \mathbb {R}$$ with a new topological invariant, $$\textit {WR}(V_{{\mathbb {C}}})$$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $$V$$. This invariant lies between $$W(V)$$ and the group $$KO(V_{ \mathbb {R}})$$ of $$ \mathbb {R}$$-linear topological vector bundles on the space $$V_{ \mathbb {R}}$$ of real points of $$V$$.

We show that the comparison maps $$W(V)\to \textit {WR}(V_{{\mathbb {C}}})$$ and $$\textit {WR}(V_{{\mathbb {C}}})\to KO(V_{ \mathbb {R}})$$ are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for the exponent of the kernel and cokernel, depending upon the dimension of $$V.$$ These results improve theorems of Knebusch, Mahé and Brumfiel.

Along the way, we prove the comparison theorem between algebraic and topological Hermitian $$K$$-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.

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