Homomorphisms of fundamental groups of planar continua

Abstract

We prove that every homomorphism from the fundamental group of a planar Peano continuum to the fundamental group of a planar or one-dimensional Peano continuum is induced by a continuous map up to conjugation. This is then used to provide a family of uncountable many planar Peano continua with pairwise non-isomorphic fundamental groups all of which are not homotopy equivalent to a one-dimensional space.of any one-dimensional Peano continuum.Proof. Let {U 1 , U 2 , · · · } be a countable set of disjoint open subsets of (0, 1)×(0, 1). In each U i we can find a subset X i such that X i ⊂ S and X i is homeomorphic to the wedge of i-closed intervals.ForĀ ⊂ N, let XĀ = i∈Ā X i . It is a trivial exercise to show that that XĀ is homeomorphic to XB if and only ifĀ =B.ForĀ ⊂ N, choose A ⊂ N such that K(S A ) = XĀ. The corollary then follows from Proposition 2.14.

Scattered products in fundamental groupoids

“…g n for all k ∈ N. For example, the existence and uniqueness of classical infinite sums and products in R depends on the topology of R. Non-abelian groups and groupoids with infinitary operations arise naturally in the context of "wild" algebraic topology, in particular, the study of fundamental group(oid)s of spaces with non-trivial local homotopy. The welldefinedness of such products in fundamental group(oid)s of one-dimensional and planar sets plays an important role in Katsuya Eda's homotopy classification of one-dimensional Peano continua [15] and related "automatic continuity" results for fundamental groups of one-dimensional and planar Peano continua [8,12,22].…”

“…The use of such infinite products is ubiquitous in the progressive literature on the homotopy and homology groups of such spaces, e.g. [6,8,9,13,16,23]. For example, fundamental group(oid)s of one-dimensional spaces [14,15,17,21] are very much infinitary extensions of fundamental group(oid)s of graphs since path-homotopy classes have "reduced" representatives that are unique up to reparameterization [5].…”

On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

“…When working with spaces for which homomorphisms between fundamental groups are induced by continuous maps up to base point change, as is the case among all one‐dimensional and planar Peano continua [8, 15, 25], the following provides additional utility: Corollary Suppose both and have the discrete monodromy property. Let be an isomorphism with and for some maps and and some paths and .…”

On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

“…When working with spaces for which homomorphisms between fundamental groups are induced by continuous maps up to base point change, as is the case among all one-dimensional and planar Peano continua [8,15,23], the following provides additional utility: Corollary 9.19. Suppose both X and Y have the discrete monodromy property.…”