Aspherical space

In <a href="/facts/Topology/2EqW47cU">topology</a>, a branch of mathematics, an aspherical space is a <a href="/facts/Path_connected/5CUTxfoA">path connected</a> <a href="/facts/Topological_space/v2ut7CbK">topological space</a> with all <a href="/facts/Homotopy_groups/xjogL95Y">homotopy groups</a> 
 
 
 
 
 π
 
 n
 
 
 (
 X
 )
 
 
 {\displaystyle \pi _{n}(X)}
 
 equal to 0 when 
 
 
 
 n
 ≠
 1
 
 
 {\displaystyle n\not =1}
 
.
If one works with <a href="/facts/CW_complex/rN7buLMo">CW complexes</a>, one can reformulate this condition: an aspherical CW complex is a CW complex whose <a href="/facts/Universal_cover/bDRGasl7">universal cover</a> is <a href="/facts/Contractible/REXYj8uS">contractible</a>. Indeed, contractibility of a universal cover is the same, by <a href="/facts/Whitehead%2527s_theorem/43UKropG">Whitehead's theorem</a>, as asphericality of it. And it is an application of the <a href="/facts/Exact_sequence_of_a_fibration/xjogL95Y">exact sequence of a fibration</a> that higher homotopy groups of a space and its universal cover are same. (By the same argument, if E is a <a href="/facts/Connected_space/5CUTxfoA">path-connected space</a> and 
 
 
 
 p
 :
 E
 →
 B
 
 
 {\displaystyle p\colon E\to B}
 
 is any <a href="/facts/Covering_space/bDRGasl7">covering map</a>, then E is aspherical if and only if B is aspherical.)
Each aspherical space X is, by definition, an <a href="/facts/Eilenberg%25E2%2580%2593MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a> of type 
 
 
 
 K
 (
 G
 ,
 1
 )
 
 
 {\displaystyle K(G,1)}
 
, where 
 
 
 
 G
 =
 
 π
 
 1
 
 
 (
 X
 )
 
 
 {\displaystyle G=\pi _{1}(X)}
 
 is the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of X. Also directly from the definition, an aspherical space is a <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> for its fundamental group (considered to be a <a href="/facts/Topological_group/cAgNMMRU">topological group</a> when endowed with the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a>).

Aspherical space open-in-new

Aspherical space