Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T14:37:48.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Pattern-equivariant cohomology with integer coefficients

Published online by Cambridge University Press:  01 December 2007

LORENZO SADUN
LORENZO SADUN*
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We relate Kellendonk and Putnam’s pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with values in any Abelian group. It also provides an easy proof of Kellendonk and Putnam’s original theorem relating PE cohomology to the Čech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1991 - 1998
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C *-algebras. Ergod. Th. & Dynam. Sys. 18 (1998), 509–537.CrossRefGoogle Scholar
[2]Bellissard, J., Benedetti, R. and Gambaudo, J.-M.. Spaces of tilings, finite telescopic approximations and gap-labeling. Comm. Math. Phys. 261 (2006), 1–41.CrossRefGoogle Scholar
[3]Benedetti, R. and Gambaudo, G.-M.. On the dynamics of G-solenoids. Applications to Delone sets. Ergod. Th. & Dynam. Sys. 23 (2003), 673–691.CrossRefGoogle Scholar
[4]Bott, R. and Tu, L.. Differential forms in algebraic topology, vol. 82 (Graduate Texts in Mathematics). Springer, New York, 1982.CrossRefGoogle Scholar
[5]Geller, W. and Propp, J.. The projective fundamental group of a -shift. Ergod. Th. & Dynam. Sys. 15 (1995), 1091–1118.CrossRefGoogle Scholar
[6]Gähler, F.. Talk given at the Aperiodic Order, Dynamical Systems, Operator Algebras and Topology Conference, 2002. Slides available at www.pims.math.ca/science/2002/adot.Google Scholar
[7]Kellendonk, J.. Pattern-equivariant functions and cohomology. J. Phys. A: Math. Gen. 36 (2003), 5765–5772.CrossRefGoogle Scholar
[8]Kellendonk, J. and Putnam, I.. The Ruelle-Sullivan map for R n actions. Math. Ann. 334 (2006).CrossRefGoogle Scholar
[9]Ormes, N., Radin, C. and Sadun, L.. A homeomorphism invariant for substitution tiling spaces. Geom. Dedicata 90 (2002), 153–182.CrossRefGoogle Scholar
[10]Priebe, N.. Towards a characterization of self-similar tilings in terms of derived Voronoi tesselations. Geom. Dedicata 79 (2000), 239–265.CrossRefGoogle Scholar
[11]Spivak, M.. Calculus on Manifolds. Addison-Wesley, Reading, MA, 1965.Google Scholar
[12]Sadun, L.. Tiling spaces are inverse limits. J. Math. Phys. 44 (2003), 5410–5414.CrossRefGoogle Scholar
[13]Schmidt, K.. Tilings, fundamental cocycles and fundamental groups of symbolic Z d-actions. Ergod. Th. & Dynam. Sys. 18 (1998), 1473–1525.CrossRefGoogle Scholar
[14]Williams, R. F.. Expanding attractors. Publ. Math. 43 (1974), 169–203.CrossRefGoogle Scholar