Date: Fri, 15 Jan 1999 07:53:05 +0900 From: Andrzej Kozlowski Subject: Re: response to nerve query As Allen Hatcher pointed out to me, there is a misprint in Segal's statement quoted by me below, which makes it look like a statement about the nerve of a covering but in fact it somewhat different (this shows again one should really read the proofs of anything one is going to quote!). Segal's statement does not require the covering to be one by contractible sets. However, in the case of a such a covering it is easy to see that the desired result follows at once from Segal's. On Thu, Jan 14, 1999, DON DAVIS wrote: >Date: Thu, 14 Jan 1999 18:13:21 +0900 >From: Andrzej Kozlowski >Subject: Re: nerve of covering > >Actually a general case is in the literature: see G.B. Segal "Classifying >spaces and spectral sequences" Pub. Math. I.H.E.S. 1968 p. 108, >Proposition 4.1. > > >On Tue, Jan 12, 1999, DON DAVIS wrote: > >>From: JURAJ_LORINC@nbs.sk >>Date: Tue, 12 Jan 1999 15:31:24 +0200 >>Subject: Nerve of covering and homotopy type >> >> >>Hello, >> >>I am student and I'd like to ask question concerning objects in the title. >> >>Studying an article "Yuliy M. Baryshnikov: Unifying Impossibility Theorems: >>A Topological Approach, Adv. Appl. Math 14, 404-415 (1993)", I came to the >>following situation: >> >>++++++++++++++++++++++++++++++++ >>Let M=R^^n \ Delta, where Delta is diagonal in R^^n. >>For all 0>is in U_ij iff x_i >x_j. >>{U_ij|0> >>All sets U_ij are convex subsets of R^^n, their intersections are convex as >>well, and the covering of M is fine, it means that all sets and their >>intersections are contractible. >>++++++++++++++++++++++++++++++++ >> >>Baryshnikov asserts that then the nerve of covering (N_M) has the same >>homotopy type as M itself. I don't see why. The strongest thing I was able >>to prove using literature, is that M and N_M have the same cohomologies. >> >>Can anybody give an exact argument for Baryshnikov's assertion? >> >>Best wishes, >> >> JUraj Lorinc >> juraj.lorinc@nbs.sk >> >> >> >> > > >Andrzej Kozlowski >Toyama International University >JAPAN >http://sigma.tuins.ac.jp/ >http://eri2.tuins.ac.jp/ > > Andrzej Kozlowski Toyama International University JAPAN http://sigma.tuins.ac.jp/ http://eri2.tuins.ac.jp/