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