Date: Thu, 14 Jan 1999 18:13:21 +0900
From: Andrzej Kozlowski <andrzej@tuins.ac.jp>
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 <dmd1@Lehigh.EDU> 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<i, j<=n, i not equal j, U_ij is hemispace in M, (x_1, ..., x_n)
>is in U_ij iff x_i >x_j.
>{U_ij|0<i, j<=n, i not equal j} is covering of M, denote his nerve N_M.
>
>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/