# '22

36th Summer Topology Conference

July 18-22, 2022

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

### "$K$-theory of two-dimensional substitution tiling spaces from $AF$-algebras"

#### Liu, Jianlong

A tiling space carries an action by translations, giving a groupoid $C^\ast$-algebra. If the tiling arises from a substitution rule, one obtains an $AF$-algebra for each intermediate dimension. For $d=1$, Putnam (1989) proved that these $AF$-algebras are sufficient in constructing the $K$-theory of the groupoid $C^\ast$-algebra. For $d=2$, Julien-Savinien (2016) showed a proof-of-concept for the chair tiling. Using the six-term exact sequence in relative $K$-theory introduced by Haslehurst (2021), we give a simple proof for $d=1,2$ that the $K$-theory of this groupoid $C^\ast$-algebra can always be constructed from that of the attached $AF$-algebras, and, incidentally, relate everything back to cohomology.

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