Abstract
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.
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Notes
- 1.
ANR = “Absolute Neighborhood Retract.” For finite-dimensional spaces this is equivalent to “locally contractible,” and is used to rule out local point-set pathology.
- 2.
There are “exotic” homology manifolds not equivalent to topological manifolds, [BFMW] but they are extremely difficult to construct and are not produced by any known natural process.
- 3.
There is a technical definition of “ghastly” in [davermanwalsh] that lives up to the name.
- 4.
This is, in fact, Poincaré’s picture of duality, and will be discussed further in the history section.
- 5.
The spaces \(V\) are vector spaces and \(\ell\) linear because we are assuming \(b_{1} = 0\) (homology sphere). The general situation is more complicated.
- 6.
The finiteness hypothesis on \(X\) is missing in the statement in [mano13].
- 7.
Blocks can be modified to eliminate the transient boundary [rotvdh], but it is best not to make this part of the definition because it makes the “plug” variation hard to formulate.
- 8.
The \(Pin(2)\) symmetry was observed much earlier, cf., [bauerFuruta], but not fully exploited.
- 9.
We now think of a topological space as a structure (a topology) on a set. In the Poincaré tradition, spaces were primitive objects with properties extrapolated from those of subsets of Euclidean spaces.
- 10.
The generalized Poincaré conjecture is the assertion that a polyhedron that is known to be a PL manifold and that has the homotopy type of the sphere is PL equivalent to the sphere.
- 11.
In 1941, after the dust had settled, J. H. C. Whitehead reviewed the various proposals from the 1920s. Brouwer’s proposal was particularly dysfunctional, and one has to wonder if he had actually tried to work with it in any serious way.
- 12.
Existence of piecewise-smooth triangulations was shown in 1940 by Whitehead [whitehead40].
- 13.
Milnor’s discovery also invalidated the intuition, inherited from Poincaré, that there would be a single world of “manifolds” where all techniques would be available. Subsequent developments, as we have seen here, revealed how confining that intuition had been.
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This work was partially supported by the Max Planck Institute for Mathematics, in Bonn.
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Quinn, F. (2016). The Triangulation of Manifolds: Topology, Gauge Theory, and History. In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds) Arbeitstagung Bonn 2013. Progress in Mathematics, vol 319. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43648-7_11
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