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Fullerenes, polytopes and toric topology. (English) Zbl 1414.52006

Darby, Alastair (ed.) et al., Combinatorial and Toric homotopy. Introductory lectures. Based on the program “Combinatorial and Toric Homotopy”, held at the National University of Singapore’s Institute for Mathematical Sciences (IMS), August 1–31, 2015. Hackensack, NJ: World Scientific. Lect. Notes Ser., Inst. Math. Sci., Natl. Univ. Singap. 35, 67-178 (2018).
The lectures deal with the combinatorial study of fullerenes. These are 3-dimensional convex polytopes arising in “quantum physics, quantum chemistry and nanotechnology”. The techniques range from particular constructions for 3-dimensional polytopes over moment-angle manifolds to quasitoric manifolds. Many nice figures illustrate the introduced concepts. Many proofs are omitted and a reference is given instead.
The content is structured in the following way. §1 consists mainly of a “lecture guide”. §2 gives an extensive introduction to three-dimensional polytopes. §3 specializes on simple polytopes. It shows some important constructions for realizing given sequences of vertex degrees. This is applied to a particular class of simple 3-polytopes, combinatorial fullerenes, in §4. From this, first characterizing properties of fullerenes are derived. To start a deeper investigation of these objects, §5 gives an overview of the tools from toric topology, in particular moment-angle complexes and moment-angle manifolds. The reader is referred to [the first author and T. Panov, Toric topology. Providence, RI: American Mathematical Society (2015; Zbl 1375.14001)] for more details. §6 deals with the cohomology groups of moment-angle complexes and moment-angle manifolds. This is applied to three-dimensional polytopes in §7. In particular, using the results from §4, further combinatorial properties of fullerenes are deduced. The rigidity of the constructions from §4 is studied in §8. Quasitoric manifolds are briefly introduced in §9, and their cohomology ring for a fullerene is identified. The lecture ends with further constructions of fullerenes in §10.
For the entire collection see [Zbl 1382.55001].

MSC:

52B10 Three-dimensional polytopes
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
57S10 Compact groups of homeomorphisms
14M25 Toric varieties, Newton polyhedra, Okounkov bodies

Citations:

Zbl 1375.14001
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