The Plus Construction

Abstract

For any associative ring R,we regard GL(R) as a topological group with the discrete topology, and let BGL(R) denote the ‘classifying space’ of GL(R). For our purposes, it is only important to know that BGL(R) is an Eilenberg–MacLane space K(GL(R)),1), i.e., BGL(R) is a connected space with π₁(BGL(R)) ≅ GL(R), π _i (BGL(R)) = 0 for i ≥ 2, and that these properties characterize BGL(R) up to homotopy equivalence (since we are assuming that all spaces considered here have the homotopy type of a CW-complex). We give a construction of the classifying space of a discrete group in the next chapter (Example (3.10)).

In this section, all spaces, pairs, etc. have the homotopy type of a CW-complex. The main reference for this chapter is: J.-L. Loday, K-théorie algébrique et représentations de groupes, Ann. Sci. Ecole Norm. Sup. 9 (1976). A general reference for topology is: G.W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math., No. 61, Springer-Verlag. These are cited below as [L] and [W], respectively.