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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
1
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The Symmetry of a Soccer Ball
An interesting example involving self-intersection, but the valence is 4 is given on page 158 of Wenninger's Polyhedron Models (Model #102), this is Great Dodecahemicosahedron, with 10 hexagons and 12 …
4
votes
Database of polyhedra
If you are interested in abstract polyhedra you can consult the atlas of small regular polytopes put together by Michael Hartley, however some of the links seem to be broken right now (I'll follow up …