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