A homological fibration for \(Gl_\infty\)
The goal of this paper is to provide a new tool for the computation of the integral homology of the infinite general linear groups \(GL(F)=\lim_nGL_n(F)\) over an infinite field \(F\). Of course, the knowledge of their homology would imply nice consequences in algebraic K-theory. More precisely, the paper is devoted to the comparison of \(H_*(GL(E);\mathbb Z)\) and \(H_*(GL(F);\mathbb Z)\) for a finite degree extension of infinite fields \(E/F\). By using Suslin's stability result \(H_i(GL_n(F);\mathbb Z)\cong H_i(GL_{n+1}(F);\mathbb Z)\), whenever \(n\geq i\), for any infinite field \(F\) [see \textit{A. A. Suslin}, Lect. Notes Math. 1046, 357-375 (1984; Zbl 0528.18007)], the author constructs a fibration in which the fiber, respectively the total space and the base space, have the same integral homology as the classifying space \(BGL(F)\), respectively \(BGL(E)\) and \(X/GL(E)\), where \(X\) is an acyclic complex with \(GL(E)\)-action. Consequently, the main result of the paper asserts the existence of a spectral sequence \(E^2_{p,q}\cong H_p(X/GL(E);H_q(GL(F);\mathbb Z))\Rightarrow H_{p+q}(GL(E);\mathbb Z)\).
scientific article; zbMATH DE number 1014011
scientific article