Algebraic & Geometric Topology Volume 26, issue 3 (2026)

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ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

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New results on tilings via cup products and Chern characters on tiling spaces

Jianlong Liu, Jonathan Rosenberg and Rodrigo Treviño

Algebraic & Geometric Topology 26 (2026) 1155–1193

DOI: 10.2140/agt.2026.26.1155

Bibliography

1	E Akkermans, Y Don, J Rosenberg, C L Schochet, Relating diffraction and spectral data of aperiodic tilings : towards a Bloch theorem, J. Geom. Phys. 165 (2021) 104217 MR4236456
2	J E Anderson, I F Putnam, Topological invariants for substitution tilings and their associated C^∗-algebras, Ergodic Theory Dynam. Systems 18 (1998) 509 MR1631708
3	M F Atiyah, G B Segal, The index of elliptic operators, II, Ann. of Math. (2) 87 (1968) 531 MR236951
4	M Baake, U Grimm, Squirals and beyond : substitution tilings with singular continuous spectrum, Ergodic Theory Dynam. Systems 34 (2014) 1077 MR3227148
5	M Barge, B Diamond, Cohomology in one-dimensional substitution tiling spaces, Proc. Amer. Math. Soc. 136 (2008) 2183 MR2383524
6	M Barge, B Diamond, J Hunton, L Sadun, Cohomology of substitution tiling spaces, Ergodic Theory Dynam. Systems 30 (2010) 1607 MR2736888
7	J Bellissard, K-theory of C^∗-algebras in solid state physics, from: "Statistical mechanics and field theory: mathematical aspects" (editors T C Dorlas, N M Hugenholtz, M Winnink), Lecture Notes in Phys. 257, Springer (1986) 99 MR862832
8	J Bellissard, R Benedetti, J M Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys. 261 (2006) 1 MR2193205
9	M T Benameur, V Mathai, Proof of the magnetic gap-labelling conjecture for principal solenoidal tori, J. Funct. Anal. 278 (2020) 108323 MR4030282
10	M T Benameur, H Oyono-Oyono, Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard), from: "Operator algebras and mathematical physics" (editors J M Combes, J Cuntz, G A Elliott, G Nenciu, H Siedentop, Ş Strătilă), Theta (2003) 11 MR2018220
11	B Blackadar, K-theory for operator algebras, 5, Cambridge Univ. Press (1998) MR1656031
12	G E Bredon, Equivariant cohomology theories, 34, Springer (1967) MR214062
13	F Gähler, J Hunton, G Maloney, A modified dual-tiling approach to the Anderson–Putnam method, lecture slides (2013)
14	A Hatcher, Vector bundles and K–theory, preprint (2017)
15	T Kaczynski, M Mrozek, The cubical cohomology ring : an algorithmic approach, Found. Comput. Math. 13 (2013) 789 MR3105945
16	J Kaminker, I Putnam, A proof of the gap labeling conjecture, Michigan Math. J. 51 (2003) 537 MR2021006
17	J Kellendonk, I F Putnam, Tilings, C^∗-algebras, and K-theory, from: "Directions in mathematical quasicrystals" (editors M Baake, R V Moody), CRM Monogr. Ser. 13, Amer. Math. Soc. (2000) 177 MR1798993
18	J Kellendonk, I F Putnam, The Ruelle–Sullivan map for actions of ℝⁿ, Math. Ann. 334 (2006) 693 MR2207880
19	D Lind, B Marcus, An introduction to symbolic dynamics and coding, Cambridge Univ. Press (2021) MR4412543
20	W Lück, B Oliver, Chern characters for the equivariant K-theory of proper G-CW-complexes, from: "Cohomological methods in homotopy theory" (editors J Aguadé, C Broto, C Casacuberta), Progr. Math. 196, Birkhäuser (2001) 217 MR1851256
21	I G Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319 MR151460
22	S Mardešić, J Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971) 41 MR298634
23	H Moustafa, PV cohomology of the pinwheel tilings, their integer group of coinvariants and gap-labeling, Comm. Math. Phys. 298 (2010) 369 MR2669441
24	N Ormes, C Radin, L Sadun, A homeomorphism invariant for substitution tiling spaces, Geom. Dedicata 90 (2002) 153 MR1898159
25	J Rosenberg, The Künneth theorem in equivariant K-theory for actions of a cyclic group of order 2, Algebr. Geom. Topol. 13 (2013) 1225 MR3044609
26	J Rosenberg, Complex K-theory of 4-complexes, (2024) arXiv:2409.16557
27	L Sadun, Topology of tiling spaces, 46, Amer. Math. Soc. (2008) MR2446623
28	G Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. (1968) 129 MR234452
29	G Segal, The representation ring of a compact Lie group, Inst. Hautes Études Sci. Publ. Math. (1968) 113 MR248277
30	B Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997) 695 MR1452190
31	B Solomyak, R Treviño, Spectral cocycle for substitution tilings, Ergodic Theory Dynam. Systems 44 (2024) 1629 MR4741328
32	C Starling, K-theory of crossed products of tiling C^∗-algebras by rotation groups, Comm. Math. Phys. 334 (2015) 301 MR3304277
33	M F Whittaker, C^∗-algebras of tilings with infinite rotational symmetry, J. Operator Theory 64 (2010) 299 MR2718945