Andreas Zastrow

The fundamental groups of subsets of closed surfaces inject into their first shape groups

We show that for every subset X of a closed surface M 2 and every x0 ∈ X, the natural homomorphism ϕ : π1(X, x0) → y π1(X, x0), from the fundamental group to the first shape homotopy group, is injective. In particular, if X ( M 2 is a proper compact subset, then π1(X, x0) is isomorphic to a subgroup of the limit of an inverse sequence of finitely generated free groups; it is therefore locally free, fully residually free and residually finite. AMS Classification 55Q52, 55Q07, 57N05; 20E25, 20E26

One-dimensional sets and planar sets are aspherical

Abstract We give a relatively short proof of the theorem that planar sets are aspherical. The first proof of this theorem, by third author Andreas Zastrow, was considerably longer.

On semilocally simply connected spaces

Abstract The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces. While the condition “semilocally simply connected” plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition “homotopically Hausdorff” instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.

On small homotopies of loops

Abstract Two natural questions are answered in the negative: • “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?” • “Can adding arcs to a space cause an essential curve to become nulhomotopic?” The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π 1 - shape injective .

On Mincʼs sheltered middle path

Abstract This paper shows that a construction, which was introduced by Piotr Minc in connection with a problem that came from Helly type theorems and that allows to replace three PL-arcs with a “sheltered middle path”, can in the case of general (non-PL) paths result in the topologistʼs sine curve.

Embeddability of multiple cones

Abstract The main result of this paper is that if X is a Peano continuum such that its n th cone C n ( X ) embeds into R n + 2 then X embeds into S 2 . This solves a problem proposed by W. Rosicki.