\documentclass[12pt,leqno]{article} \usepackage{amsmath} %\usepackage{amssymb} \usepackage{amscd} \usepackage[psamsfonts]{amssymb} \parindent=0cm \begin{document} \pagestyle{empty} \textsc{Title:} Sturm sequences and $\mathrm{H}_{2}$ of the hyperbolic homomorphism \vspace{1cm} \textsc{Abstract:} Let $R$ be an arbitrary ring (let's say commutative). Let $\mathrm{GL}(R)$ and $\mathrm{Sp}(R)$ be the ``infinite" general linear group and symplectic group ; let $\mathrm{H}:\mathrm{GL}(R)\to \mathrm{Sp}(R)$ be the hyperbolic homomorphism. Finally let $\mathrm{I}(R)$ be the fundamental ideal of the Witt ring of $R$ (the one of the theory of non-degenerate symmetric bilinear forms, defined here in terms of free $R$-modules). Our aim is to show that the homology group $\mathrm{H}_{2}(\mathrm{H})$, group which takes place naturally in a five term exact sequence $$ \begin{CD} \mathrm{K}_{2}(R) @>>> \mathrm{KSp}_{2}(R) @>>> \mathrm{H}_{2}(\mathrm{H}) @>>> \mathrm{K}_{1}(R) @>>> \mathrm{KSp}_{1}(R) \hspace{8pt}, \end{CD} $$ is the fibre product of $\mathrm{I}(R)$ and $\mathrm{K}_{1}(R)$ over $\mathrm{K}_{1}(R)/(1+\tau)$, $\tau$ denoting the involution of $\mathrm{K}_{1}(R)$ defined by matrix transposition. \medskip This result is a variant of results of R. W. Sharpe [On the structure of the unitary Steinberg group, \textit{Ann. of Math.}, \textbf{96} (1972), 444-479]. Our method of proof is distantly related to the classical theory of Sturm sequences, hence our title. \end{document}