manifolds and cobordisms
cobordism theory, Introduction
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homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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For a manifold,
a simplicial triangulation is a simplicial complex and a homeomorphism from its geometric realization to the underlying topological space of .
a combinatorial triangulation is a simplicial triangulation such that for each simplex the link of (the union of all simplices that intersect such that both and are faces of a simplex) is homeomorphic to a sphere.
A triangulation conjecture conjectures and triangulation theorem proves that a given kind of triangulation exists for a given kind of manifold. Conversely, deep theorems assert that a given kind of triangulation does not generally exists for a given class of manifolds.
It is easy to show that every piecewise-linear manifold admits a combinatorial triangulation, so that combinatorial triangulability is often understood to mean existence of a piecewise-linear structure.
A compatible piecewise-linear structure, hence a combinatorial triangulation, hence a simplicial triangulation, does exist for
every smooth manifold
every topological manifold of dimension 2, hence every surface
every topological manifold of dimension 3, hence every 3-manifold
In each dimension 4 there exist topological manifolds not admitting a combinatorial triangulation (Kirby & Siebenmann 1969).
In each dimension 5 there exist topological manifolds not even admitting a simplicial triangulation (Manolescu 2016)
For topological manifolds of dimension triangulations still exist in general, but for every dimension there exist topological manifolds which do not admit a triangulation. For there is an obstruction class in the ordinary cohomology with coefficients in , in that admits a triangulation if and only if .
On triangulation conjectures and triangulation theorems on existence of triangulations of manifolds.
Review:
The question of triangulability of smooth manifolds was first raised in
and for general topological manifolds in
Proof that every surface admits a combinatorial triangulation:
Proof that every smooth manifold admits a combinatorial triangulation:
Proof that every smooth manifold admits a combinatorial triangulation is due to
Stewart S. Cairns, Triangulation of the manifold of class one, Bull. Amer. Math. Soc. 41(8): 549-552 (euclid:1183498332)
J. H. C. Whitehead, On -complexes, Ann. of Math. (2) 41 (1940), 809–824 (doi:10.2307/1968861, jstor:1968861)
with further accounts in:
Hassler Whitney, Section IV.B of: Geometric Integration Theory (1957), Princeton Legacy Library (2016) (doi:10.1515/9781400877577)
Stewart S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc. 67 (1961), 389-390 (doi:10.1090/S0002-9904-1961-10631-9)
(but see MO:a/177199, where this “simpler argument” is claimed to be wrong)
Jacob Lurie, Existence of Triangulations (pdf), Lecture 4 in: Topics in Geometric Topology
A detailed exposition is available in Chapter II (see Thm. 10.6) of
Generalization to existence of equivariant triangulation for smooth G-manifolds (equivariant triangulation theorem):
Sören Illman, Equivariant algebraic topology, Princeton University 1972 (pdf)
Sören Illman, Smooth equivariant triangulations of -manifolds for a finite group, Math. Ann. (1978) 233: 199 (doi:10.1007/BF01405351)
Sören Illman, The Equivariant Triangulation Theorem for Actions of Compact Lie Groups, Mathematische Annalen (1983) Volume: 262, page 487-502 (dml:163720)
Proof that every 3-manifold admits the structure of a smooth manifold and hence of a combinatorial triangulation:
Proof that in every dimension there exist topological manifolds without combinatorial triangulation:
Proof that in every dimension there exist topological manifolds without simplicial triangulation:
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