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The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by Henri Poincaré. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group.
May 31, 2017 · Poincaré used the fundamental group of the homology sphere to show that it was topologically different from a sphere. The Poincaré conjecture was one of the most...
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The Poincaré conjecture was a mathematical problem in the field of geometric topology. In terms of the vocabulary of that field, it says the following: Poincaré conjecture. Every three-dimensional topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic to the three-dimensional sphere.
Poincaré's homology sphere is a closed 3- manifold with the same homology as the 3-sphere but with a fundamental group which is non-trivial. In his series of papers on Analysis situs (1892 - 1904) Poincaré introduced the fundamental group and studied Betti-numbers and torsion coefficients.
3 days ago · Homology is a concept that is used in many branches of algebra and topology. Historically, the term "homology" was first used in a topological sense by Poincaré. To him, it meant pretty much what is now called a bordism , meaning that a homology was thought of as a relation between manifolds mapped into a manifold.
We will explain how complex hypersurface singularities give rise to knots and links, some of which are manifolds that are homologically spheres but not topologically (Poincar e homology spheres), or manifolds that are homeomorphic to spheres but not di eomorphic to them (exotic spheres).
Apr 25, 2022 · Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. Homology of a polyhedron ), but the axiomatic construction (including the precise limits of this concept, which had been indefinite for a long time) was imparted to it only by S. Eilenberg and N. Steenrod [3] (cf. Algebraic topology; Homology group; Steenrod–E...
