In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups equal to 0 when .
If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is contractible. Indeed, contractibility of a universal cover is the same, by Whitehead's theorem, as asphericality of it. And it is an application of the exact sequence of a fibration that higher homotopy groups of a space and its universal cover are same. (By the same argument, if E is a path-connected space and is any covering map, then E is aspherical if and only if B is aspherical.)
Each aspherical space X is, by definition, an Eilenberg–MacLane space of type , where is the fundamental group of X. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group when endowed with the discrete topology).
space and p : E → B {\displaystyle p\colon E\to B} is any covering map, then E is aspherical if and only if B is aspherical.) Each asphericalspace X...
Aspherical may refer to: Asphericalspace, a concept in topology Aspherical lens, a type of lens assembly used in photography which contains an aspheric...
galactic dark-matter haloes was titled "Milking the spherical cow – on aspherical dynamics in spherical coordinates". References to the joke appear even...
interpretation. It is known that H {\displaystyle \mathbb {H} } is an asphericalspace, i.e. all higher homotopy and homology groups of H {\displaystyle \mathbb...
spaces, such as tori, all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called asphericalspaces....
determined up to isomorphism by their fundamental group. The total spaces are aspherical (in other words all higher homotopy groups vanish). They have Thurston...
fibration is a proper approximate fibration between manifolds. asphericalspaceAsphericalspace assembly map Atiyah 1. Michael Atiyah. 2. Atiyah duality...
solvmanifold is aspherical. Among all compact homogeneous spaces, solvmanifolds may be characterized by the properties of being aspherical and having a solvable...
well as whether a best spherical form of lens or an optically optimal aspherical form was used in the manufacture of the lens. Generally, the best spherical...
the assembly map in L-theory. The Borel conjecture on the rigidity of aspherical manifolds is equivalent to the assembly map being an isomorphism. Davis...
important effect, because spherical shapes are much easier to produce than aspherical ones. In many cases, it is cheaper to use multiple spherical elements...
homeomorphism. Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds...
optical components are an easy-to-make spherical primary mirror, and an aspherical correcting lens, known as a Schmidt corrector plate, located at the center...
Eilenberg–MacLane space of the finite cyclic group of order 2. All compact aspherical manifolds are essential (since being aspherical means the manifold...
is about 5000 km². The particles in the main ring are expected to have aspherical shapes. The total mass of the dust is estimated to be 107−109 kg. The...
glass.[citation needed] Plastics allow the manufacturing of strongly aspherical lens elements which are difficult or impossible to manufacture in glass...
}\mathbb {Z} \oplus \oplus _{i=1}^{\lfloor n/2\rfloor }\mathbb {Z} _{2}} An aspherical n {\displaystyle n} -dimensional manifold X {\displaystyle X} is an n...
sign conjecture but which does not refer to Riemannian geometry at all. Aspherical manifolds are connected manifolds for which all higher homotopy groups...
glass element, and/or certain types of aspherical elements. (Note that a number of non-L lenses also use aspherical elements, and at least one non-L lens...
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