2,253 research outputs found
Beyond Sperner's lemma
The present paper is devoted to a recent beautiful and ingenious proof of
Brouwer's fixed point theorem due to mathematical economists H. Petri and M.
Voorneveld. The heart of this proof is an analogue of Sperner's lemma motivated
by Shapley-Scarf model of markets of agents with preferences over indivisible
goods.
The goal of the paper is to present a relatively abstract version of
Petri-Voorneveld proof which makes transparent both its similarities and its
differences with the classical proof based on Sperner's lemma and a well known
Knaster-Kuratowski-Mazurkiewich argument.Comment: 9 page
The virtual cohomology dimension of Teichm\"uller modular groups: the first results and a road not taken
The paper is devoted to a detailed exposition of results outlined in author's
1983 note "On the virtual cohomology dimension of the Teichm\"uller modular
group". The paper includes the full proofs and a discussion of the context
which motivated both the results and the methods used. The reminiscences
included into the last section are an integral part of this discussion. A key
idea is to use the simply-connectedness of the Hatcher-Thurston complex. This
idea was hardly developed further, but still keeps a promise.Comment: 30 pages, 7 figure
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