We present here a method for computing the homology of a substitution tiling space. There is a well established cohomology theory that uses simple matrix computations to determine if two tiling spaces are dierent. We will show how to compute Putnam's homology groups for these spaces using simple linear algebra. We construct a Markov Partition based on the substitution rules, and exploit the properties of this partition as a shift of finite type to construct algebraic invariants for the tiling space. These invariants form a chain complex, of which we can compute the homology. In our examples we will demonstrate an interesting duality between the cohomology and homology of these spaces. This leads to a conjecture relating the two theories to each other and we present the reasoning behind the conjecture.
Cia Chi Tung
Date of Degree
Master of Science (MS)
Mathematics and Statistics
Science, Engineering and Technology
Ford, Jeffrey Myers, "A Duality Theory for the Algebraic Invariants of Substitution Tiling Spaces" (2011). All Graduate Theses, Dissertations, and Other Capstone Projects. 179.
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License