36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"The (profinite) fundamental group of a tiling space"

Gähler, Franz

Quite generally, tiling spaces can be regarded as inverse limits of CW-complexes. As a result, these spaces are not path-connected, so that topological invariants which see only the path components will miss essential information. This is the case for homotopy groups, and the fundamental group in particular. Also, since the fundamental group is not well-behaved when taking inverse limits, it cannot be computed from the fundamental groups of the approximant complexes, as it can be done for \v{C}ech cohomology. Here, we construct a profinite version of the fundamental groups of the approximant complexes of a tiling space, and show that their inverse limit is not only well defined, but provides a genuine invariant for the inverse limit space. While such a profinite fundamental group $\hat{\pi}_1$ is still a complicated object to study, we show that certain fingerprints of it are practically computable. This is so in particular for the size of the sets $Hom(\hat{\pi}_1,G)$, with $G$ any finite group. It turns out that such fingerprints are a powerful tool to distinguish substitution tiling spaces, especially in one dimension. This is illustrated with many examples.

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