Topological manifold whose homology coincides with that of a sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is,
and
for all other i.
Therefore X is a connected space, with one non-zero higher Betti number, namely, . It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem).
A rational homology sphere is defined similarly but using homology with rational coefficients.
In algebraic topology, a homologysphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 {\displaystyle n\geq 1} ...
groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homologysphere. The topological generalized Poincaré conjecture...
optical properties of a mirror sphere Hoberman sphereHomologysphere Homotopy groups of spheres Homotopy sphere Lenart Sphere Napkin ring problem Orb (optics)...
oriented integral homology 3-spheres, introduced by Andrew Casson. Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker...
Real projective plane, RP2 Sphere, S2 Surface of genus g Torus Double torus 3-sphere, S3 3-torus, T3 Poincaré homologysphere SO(3) ≅ RP3 Solid Klein bottle...
to the higher spheres is null-homotopic. If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is...
Degree of a continuous mapping Borsuk–Ulam theorem Ham sandwich theorem Homologysphere Homotopy Path (topology) Fundamental group Homotopy group Seifert–van...
(singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful...
manifold. An example of a homology manifold that is not a manifold is the suspension of a homologysphere that is not a sphere. If X×Y is a topological...
simplicial sphere. In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homologyspheres. For any n...
double suspension S2X of a homologysphere X is a topological sphere. If X is a piecewise-linear homologysphere but not a sphere, then its double suspension...
homeomorphic to the Poincaré homologysphere. Freedman's theorem on fake 4-balls then says we can cap off this homologysphere with a fake 4-ball to obtain...
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated...
mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide...
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algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological...
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises...
topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H n ( X )...
after completing a senior thesis on Actions with fixed-point set: a homologysphere, supervised by William Browder. He received his PhD in mathematics...
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