The Witt group of real surfaces

TL;DR

This paper compares algebraic and topological Witt groups of real algebraic surfaces, developing tools to compute the topological group and analyze differences with the algebraic group.

Contribution

It introduces topological methods to compute the topological Witt group and compares it with the algebraic Witt group for real surfaces, especially in dimension two.

Findings

Developed topological tools for calculating $WR(V_{top})$

Established methods to measure differences between $W(V)$ and $WR(V_{top})$

Applied techniques specifically to 2-dimensional real algebraic varieties

Abstract

Let $V$ be an algebraic variety defined over $\mathbb R$, and $V_{top}$ the space of its complex points. We compare the algebraic Witt group $W(V)$ of symmetric bilinear forms on vector bundles over $V$, with the topological Witt group $WR(V_{top})$ of symmetric forms on Real vector bundles over $V_{top}$ in the sense of Atiyah, especially when $V$ is 2-dimensional. To do so, we develop topological tools to calculate $WR(V_{top})$, and to measure the difference between $W(V)$ and $WR(V_{top})$.