2025/10/27 by Brazas, Jeremy, Pavesic, Petar
#54F50 #55M30 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.22901
We study the Lusternik-Schnirelmann category and topological complexity of 1-dimensional spaces. We define both invariants as lengths of suitable closed filtrations, as opposed to a more common definition based on open covers. Our main results provide a precise description of cat(X) and TC(X) of a 1-dimensional Peano continuum X in terms of the wildness rank of X. A surprising consequence is that cat(X) and TC(X) of a general 1-dimensional space X can be arbitrarily high, which is in stark contrast with the analogous results for 1-dimensional CW-complexes.