The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. Minimal homeomorphisms are constructed on compact connected metric spaces with any prescribed finitely generated K-theory or cohomology. In particular, although a non-zero Euler characteristic obstructs the existence of a minimal homeomorphism on a finite CW-complex, this is not the case on a compact metric space. We also allow for some control of the map on K-theory and cohomology induced from these minimal homeomorphisms. This allows for the construction of many minimal homeomorphisms that are not homotopic to the identity. Applications to C∗-algebras will be discussed in another paper.
Robin J. Deeley
1
;
Ian F. Putnam
2
;
Karen R. Strung
3
1
University of Colorado, Boulder, USA
2
University of Victoria, Canada
3
Czech Academy of Sciences, Prague, Czechia
Robin J. Deeley; Ian F. Putnam; Karen R. Strung. Minimal homeomorphisms and topological $K$-theory. Groups, geometry, and dynamics, Tome 17 (2023) no. 2, pp. 501-532. doi: 10.4171/ggd/707
@article{10_4171_ggd_707,
author = {Robin J. Deeley and Ian F. Putnam and Karen R. Strung},
title = {Minimal homeomorphisms and topological $K$-theory},
journal = {Groups, geometry, and dynamics},
pages = {501--532},
year = {2023},
volume = {17},
number = {2},
doi = {10.4171/ggd/707},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/707/}
}
TY - JOUR
AU - Robin J. Deeley
AU - Ian F. Putnam
AU - Karen R. Strung
TI - Minimal homeomorphisms and topological $K$-theory
JO - Groups, geometry, and dynamics
PY - 2023
SP - 501
EP - 532
VL - 17
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/707/
DO - 10.4171/ggd/707
ID - 10_4171_ggd_707
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%A Ian F. Putnam
%A Karen R. Strung
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%J Groups, geometry, and dynamics
%D 2023
%P 501-532
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%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/707/
%R 10.4171/ggd/707
%F 10_4171_ggd_707
