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On the homotopy transfer of A structures

Jakub Kopřiva (2017)

Archivum Mathematicum

The present article is devoted to the study of transfers for A structures, their maps and homotopies, as developed in [7]. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.

On uncountable collections of continua and their span

Dušan Repovš, Arkadij Skopenkov, Evgenij Ščepin (1996)

Colloquium Mathematicae

We prove that if the Euclidean plane 2 contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree X 2 such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.

Optimal bounds for the colored Tverberg problem

Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler (2015)

Journal of the European Mathematical Society

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

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