Contents
Preface vii
Chapter 1. Basic notions 1
1.1. Tilings 1
1.2. Tiling spaces 6
1.3. Equivalence 8
1.4. Contructing interesting tilings 9
Chapter 2. Tiling spaces and inverse limits 21
2.1. Local structure 21
2.2. Inverse limit spaces 21
2.3. Tiling spaces are inverse limits 22
2.4. ahler’s construction 23
2.5. The Anderson-Putnam construction 25
Chapter 3. Cohomology of tilings spaces 31
3.1. Direct limits 32
3.2. Cohomology and limits 33
3.3.
ˇ
Cech cohomology of an open cover 35
3.4. Cofinal sets, good covers and open stars 36
3.5. Cohomology of inverse limits 38
3.6. Shape deformations 41
3.7. Topological conjugacy 43
Chapter 4. Relaxing the rules I: Rotations 45
4.1. A new topology for tiling spaces 46
4.2. Local structure and global topology 47
4.3. Inverse limit structures 47
4.4. Quotients of the tiling space 50
4.5. Three spaces of chair tilings 50
4.6. A flat-Earth calculation 52
4.7. Penrose cohomology 56
Chapter 5. Pattern-equivariant cohomology 61
5.1. Pattern-equivariant functions 62
5.2. How to view pattern equivariance 63
5.3. Integer coefficients 65
5.4. Interpreting tiling cohomology 66
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