The Plus Construction and Applications
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 H1(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 > O. The new complex will depend on G (as BG does) but the higher homotopy groups π i(BG +) can be highly complicated invariants of G.
KeywordsSpectral Sequence Homotopy Type Homotopy Group Poincare Series Homotopy Fiber
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