Abstract
A Hausdorff topological space X is called an n-dimensional manifold or simply an n-manifold, if for every point x ∈ X there exists an open set U of X that contains x and is homeomorphic to an open set of ℝ n. Hence, an n-manifold X is characterized by a set \(\mathfrak{A} =\{ ({U}_{i},{\phi }_{i})\ \vert \ i \in J\}\), where U i are open sets covering X, and ϕ i is a homeomorphism from U i onto an open set of ℝ n. The set \(\mathfrak{A}\) is the atlas of X and each pair (U i , ϕ i ) is a chart of X.
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Notes
- 1.
In the literature, it is sometimes required that the topological space X fulfill other conditions to be defined as a manifold (e.g., second countable, paracompact).
- 2.
See [8] for an elementary proof, based essentially on the Jordan–Schoenflies Theorem: a simple closed curve J on the Euclidean plain divides it into two regions and there exists a homeomorphism from the plane in itself that sends J into a circle.
- 3.
For instance, U may be defined as | T | together with the interior of all triangles and the sides of the second barycentric subdivision of K that intersect T.
- 4.
For a proof based on identifying polygons, see William Massey [25, Theorem 1.5.1].
- 5.
Not in the sense of our definition of 2-manifold; it is really a 2-manifold with boundary.
- 6.
We remember that our definition of triangulable manifold requires K to be connected by 1-simplexes.
- 7.
We remind the reader that c σ is the map that takes the p-simplex σ to 1 ∈ ℤ and all other p-simplexes to 0 ∈ ℤ.
- 8.
In particular, if π1(M) = 0, then M has the same homology groups as the 3-sphere S 3; in 1904, Poincaré asked the question: in this case, is it true that M is homeomorphic to S 3? It was only recently that this famous “Poincaré conjecture” was proved affirmatively by Grigory Perelman, who used methods of differential geometry, specially, the Ricci flow.
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© 2011 Springer Science+Business Media, LLC
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Ferrario, D.L., Piccinini, R.A. (2011). Triangulable Manifolds. In: Simplicial Structures in Topology. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7236-1_5
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DOI: https://doi.org/10.1007/978-1-4419-7236-1_5
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