The Plus Construction
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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), πi(BGL(R)) = 0 for i≥2, and that these properties characterise BGL(R) upto 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)).
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