Three quick responses to Stasheff question............DMD _______________________________________________ Subject: To Jim From: Peter May Date: Thu, 16 Nov 2006 09:54:49 -0600 Let c_g: G >--> G be conjugation by g, c_g(h) = g^{-1}hg. Let f: EG >--> EG be right multiplication by g, f(x) = xg. Then f(xh) = f(x)c_g(h), which means that f induces Bc_g on passage to classifying spaces. The induced map on BG is obtained by passage to orbits and is thus the identity. Peter _______________________________________________ Subject: Re: two postings From: Rainer Vogt Date: Thu, 16 Nov 2006 17:10:50 +0100 > > Subject: query > From: jim stasheff > Date: Sat, 11 Nov 2006 14:19:29 -0500 > > Consider the classifying space functor B:TopGrp --> Top > > I sem to recall that B maps conjugate maps to homotopic maps > > But my aging neurons come up with a proof only for path connected groups - the path from the conjugating element to the > identity goes over to the homotopy > > Can anyone do better? > thanks > > jim > Dear Jim, let f:G \to G be conjugation by x. Then B(f) is homotopic to the identity: Consider G as a topologically enriched category with one object. Then f is a functor and multiplication with x defines a natural transformation of f to the identity. The natural transformation becomes a homotopy after application of B. Regards Rainer Prof. Dr. Rainer Vogt Studiendekan Direktor des Instituts fuer Mathematik Fachbereich Mathematik/Informatik Albrechtstrasse 28 Osnabrueck 49076 Germany _______________________________________________________________ Subject: Response to Stasheff From: Kari Ragnarsson Date: Thu, 16 Nov 2006 10:27:01 -0600 (CST) Here's a suggestion: For groups G and H, Consider the fibre sequence Map_*(BG,BH)-> Map(BG,BH) -> BH. At the end of the LES in homotopy you get \pi_1(BH) -> \pi_0(Map_*(BG,BH)) -> \pi_0(Map(BG,BH)) -> \pi_0(BH) = * This should be read as "\pi_1(BH) = \pi_0(H) acts on \pi_0(Map_*(BG,BH)) with quotient \pi_0(Map(BG,BH))." This action of \pi_0(H) corresponds to conjugation. Now add your proof for actions within the connected component of the identity, and you should have all the ingredients. Kari