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On the Extension of Certain Maps with Values in Spheres

Carlos Biasi, Alice K. M. Libardi, Pedro L. Q. Pergher, Stanisław Spież (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let S n - 2 S be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if h : V S n - 2 is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to S n - 2 and with g - 1 ( S n - 2 ) = V . Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism...

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.

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