Aspherical Space

(Algebraic Topology)

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Algebraic Topology
Link to Dbpedia

What is Aspherical space?

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).

Technology Types

algebraic topologyhomology theoryhomotopy theory

Synonyms

Aspherical manifold

Translations

Espace asphérique (fr)Espaço asférico (pt)

Tech Info


Sources: DBpedia
 — Date merged: 11/6/2021, 1:32:53 PM
 — Date scraped: 5/20/2021, 6:04:43 PM