# Aspherical space

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In [topology](/source/topology), a branch of [mathematics](/source/mathematics), an '''aspherical space''' is a [path connected](/source/path_connected) [topological space](/source/topological_space) with all [homotopy groups](/source/homotopy_groups) <math>\pi_n(X)</math> equal to 0 when <math>n\not = 1</math>.

If one works with [CW complex](/source/CW_complex)es, one can reformulate this condition: an aspherical CW complex is a CW complex whose [universal cover](/source/universal_cover) is [contractible](/source/contractible). Indeed, contractibility of a universal cover is the same, by [Whitehead's theorem](/source/Whitehead's_theorem), as asphericality of it. And it is an application of the [exact sequence of a fibration](/source/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](/source/Connected_space) and <math>p\colon E \to B</math> is any [covering map](/source/covering_space), then ''E'' is aspherical if and only if ''B'' is aspherical.)

Each aspherical space ''X'' is, by definition, an [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) of type <math>K(G,1)</math>, where <math>G = \pi_1(X)</math> is the [fundamental group](/source/fundamental_group) of ''X''.  Also directly from the definition, an aspherical space is a  [classifying space](/source/classifying_space) for its fundamental group (considered to be a [topological group](/source/topological_group) when endowed with the [discrete topology](/source/discrete_topology)).

==Examples==
* Using the second of above definitions we easily see that all orientable compact [surface](/source/Surface_(topology))s of genus greater than 0 are aspherical (as they have either the Euclidean plane or the hyperbolic plane as a universal cover).
* It follows that all non-orientable surfaces, except the real [projective plane](/source/projective_plane), are aspherical as well, as they can be covered by an orientable surface of genus 1 or higher.
* Similarly, a [product](/source/product_topology) of any number of [circle](/source/circle)s is aspherical. As is any complete, Riemannian flat manifold.
* Any [hyperbolic 3-manifold](/source/hyperbolic_3-manifold) is, by definition, covered by the hyperbolic 3-space '''H'''<sup>3</sup>, hence aspherical. As is any ''n''-manifold whose universal covering space is hyperbolic ''n''-space '''H'''<sup>''n''</sup>.
* Let ''X''&nbsp;=&nbsp;''G''/''K'' be a [Riemannian symmetric space](/source/Riemannian_symmetric_space) of negative type, and '''Γ''' be a [lattice](/source/lattice_(discrete_subgroup)) in ''G'' that acts freely on ''X''. Then the  [locally symmetric space](/source/locally_symmetric_space) <math>\Gamma\backslash G/K</math> is aspherical.
* The [Bruhat–Tits building](/source/Bruhat%E2%80%93Tits_building) of a simple [algebraic group](/source/algebraic_group) over a field with a [discrete valuation](/source/discrete_valuation) is aspherical.
* The complement of a [knot](/source/knot_(mathematics)) in '''S'''<sup>3</sup> is aspherical, by the [sphere theorem](/source/sphere_theorem_(3-manifolds))
* Complete metric spaces with nonpositive curvature in the sense of [Aleksandr D. Aleksandrov](/source/Aleksandr_Aleksandrov_(mathematician)) (locally [CAT(0) space](/source/CAT(0)_space)s) are aspherical. In the case of [Riemannian manifold](/source/Riemannian_manifold)s, this follows from the [Cartan–Hadamard theorem](/source/Cartan%E2%80%93Hadamard_theorem), which has been generalized to [geodesic metric space](/source/geodesic_metric_space)s by [Mikhail Gromov](/source/Mikhail_Gromov_(mathematician)) and [Hans Werner Ballmann](/source/Hans_Werner_Ballmann). This class of aspherical spaces subsumes all the previously given examples.
* Any [nilmanifold](/source/nilmanifold) is aspherical.

==Symplectically aspherical manifolds==
In the context of [symplectic manifold](/source/symplectic_manifold)s, the meaning of "aspherical" is a little bit different. Specifically, we say that a symplectic manifold (M,ω) is symplectically aspherical if and only if

:<math>\int_{S^2}f^*\omega=\langle c_1(TM),f_*[S^2]\rangle=0</math>

for every continuous mapping

:<math>f\colon S^2 \to M,</math>

where <math>c_1(TM)</math> denotes the first [Chern class](/source/Chern_class) of an [almost complex structure](/source/almost_complex_manifold) which is compatible with ω.

By [Stokes' theorem](/source/Stokes'_theorem), we see that symplectic manifolds which are aspherical are also symplectically aspherical manifolds. However, there do exist symplectically aspherical manifolds which are not aspherical spaces.<ref>{{cite journal |first=Robert E. |last=Gompf |author-link=Robert Gompf|title=Symplectically aspherical manifolds with nontrivial π<sub>2</sub> |journal=Mathematical Research Letters |volume=5 |issue=5 |pages=599–603 |date=1998 |doi=10.4310/MRL.1998.v5.n5.a4 |mr=1666848 |arxiv=math/9808063 |citeseerx=10.1.1.235.9135|s2cid=15738108 }}</ref>

Some references<ref>{{cite journal |first1=Jarek |last1=Kedra |first2=Yuli |last2=Rudyak |author2-link=Yuli Rudyak |first3=Aleksey |last3=Tralle |title=Symplectically aspherical manifolds |journal=Journal of Fixed Point Theory and Applications |volume=3 |issue= |pages=1–21 |date=2008 |doi=10.1007/s11784-007-0048-z |mr=2402905 |arxiv=0709.1799 |citeseerx=10.1.1.245.455|s2cid=13630163 }}</ref> drop the requirement on ''c''<sub>1</sub> in their definition of "symplectically aspherical."   However, it is more common for symplectic manifolds satisfying only this weaker condition to be called "weakly exact."

==See also==
*[Acyclic space](/source/Acyclic_space)
*[Essential manifold](/source/Essential_manifold)
*[Whitehead conjecture](/source/Whitehead_conjecture)

==Notes==
<references/>

==References==
*{{Cite book|last1=Bridson|first1=Martin R.|author1-link=Martin Bridson|title=Metric Spaces of Non-Positive Curvature|last2=Haefliger|first2=André|author2-link=André Haefliger|year=1999|publisher=[Springer](/source/Springer_Science%2BBusiness_Media)|isbn=978-3-642-08399-0|series=Grundlehren der mathematischen Wissenschaften|volume=319|location=Berlin, Heidelberg|doi=10.1007/978-3-662-12494-9|mr=1744486}}

== External links ==
* [https://web.archive.org/web/20110719102802/http://www.map.him.uni-bonn.de/index.php/Aspherical_manifolds Aspherical manifolds] on the Manifold Atlas.

Category:Algebraic topology
Category:Homology theory
Category:Homotopy theory

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Adapted from the Wikipedia article [Aspherical space](https://en.wikipedia.org/wiki/Aspherical_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Aspherical_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.