Partially defined cocycles and the Maslov index for a local ring. (Q1774049)

A subset \(Y\subseteq G\) of a group \(G\) is called \(m\)-dense if \((g_1Y)\cap\cdots\cap(g_mY)\not=\emptyset\) for all \(g_1,\dots,g_m\in G\). Let \(Y^n_{\text{gen}}=\{(g_1,\dots,g_n)\in Y^n;\;g_i\cdots g_{i+j}\in Y\) for \(1\leq i\leq n\}\). Given an Abelian group, the author defines groups \(H^n_Y(G,B)\) for \(0\leq n\leq m-1\) to show that the embedding \(Y^n_{\text{gen}}\to G^n\) induces an isomorphism \(H^n(G,B)@>\simeq>>H^n_Y(G,B)\). At the end, let \(I(R)\) be the ideal of even forms in the Witt group \(W(R)\) of a local commutative ring \(R\). Then, via the Maslov index, the central extension \[ 0\to I^2(R)\to\widetilde{\text{Sp}(V)}\to\text{Sp}(V)\to 1 \] of the symplectic group \(\text{Sp}(V)\) by \(I^2(R)\) for a free \(R\)-module \((V,\varphi)\) with a form \(\varphi\) developed by \textit{R. Parimala, R. Preeti} and \textit{R. Sridharan} [K-Theory 19, No. 1, 29-45 (2000; Zbl 1037.11026)] is produced.