The Plus Construction and Applications
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Let G be a finite group. As we have seen, the classifying space BG has a very simple homotopy type as it is a K(G, 1). If G is perfect then H 1 (G; ℤ) = 0; suppose that we attach cells to BG to obtain a new, but simply-connected complex BG+ with the same homology as before. Or equivalently so that the homotopy fiber of BG →BG + is acyclic, i.e. H i (F; ℤ) = 0 for all i > 0. The new complex will depend on G (as BG does) but the higher homotopy groups πi(BG+) can be highly complicated invariants of G.
KeywordsExact Sequence Spectral Sequence Homotopy Type Homotopy Group Poincare Series
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