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- W2023709356 abstract "If F->-E-+B is a fibrationwith F=K(G, 1), G Abelian, and B=K(ll, 1), then it is shown that the action and characteristic class of the fibration correspond to those of the induced group extension. 1. Let F and B be spaces of the homotopy type of connected CW complexes and which are Eilenberg-Mac Lane spaces of type K(G, 1) and K(l~l, 1), respectively, where G is Abelian. Given any fibration F-VF^>Z?, it is natural to ask: what is the fundamental group of F? The only non- trivial part of the homotopy sequence for p is 0 _+ ?F) ±* irx(E) ^> MB) —* 1. Thus 7T,(F) is a group extension of trx(F) by nx(B) and F is a K(ttx(E), 1)). Since G=trx(F) is Abelian, this extension naturally induces a^Z^-module structure on G, cp:TTx(B)-^Aut(G). See (10, p. 108). The (equivalence class of this) extension is completely determined by a certain 2-dimensional group cohomology characteristic class c e ZF?(n ; G). On the other hand, in any fibration, dragging the fiber over a loop in the base space induces a homotopy equivalence on the fiber. In this way, the fibration p (geometrically) induces an action '). If ', then Hl(Yl; G) and ZZ^(Zi(II, 1); {G}) are naturally isomor- phic, and it is natural to conjecture that c corresponds to k. Indeed, this is so and in fact we will prove: Theorem 1. Let F and B be as above and F—>E—>B be a fibration. (a) The two actions ' ofTtx(B) on rrx(F) are the same." @default.
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- W2023709356 date "1973-02-01" @default.
- W2023709356 modified "2023-09-26" @default.
- W2023709356 title "On characteristic classes of groups and bundles of $K(Pi ,,1)$’s" @default.
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- W2023709356 doi "https://doi.org/10.1090/s0002-9939-1973-0319192-1" @default.
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