A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space

2022/10/22 by Brazas, Jeremy, Fischer, Hanspeter · 1 citation
#55R05 #57M10 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.12567

Abstract

We present a 2-dimensional Peano continuum \mathbbT⊆ ℝ³ with the following properties: (1) There is a universal covering projection q:\mathbbT→ \mathbbT with uncountable fundamental group π₁(\mathbbT); (2) For every 1\not=[α]∈ π₁(\mathbbT,∗), there is a covering projection r:(E,e)→ (\mathbbT,∗) such that [α]\not∈ r_#π₁(E,e); (3) There is no universal covering projection r:E→ \mathbbT; (4) The universal object p:\widetilde\mathbbT→ \mathbbT in the category of fibrations with unique path lifting (and path-connected total space) over \mathbbT has trivial fundamental group π₁(\widetilde\mathbbT)=1; (5) p:\widetilde\mathbbT→ \mathbbT is not a path component of an inverse limit of covering projections over \mathbbT.

Cited by

• Nonlocal loss of first homotopy in polyhedral approximations of Peano continua

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