Aspherical space

<h2 id="examples">Examples</h2>
<ul><li>Using the second of above definitions we easily see that all orientable compact <a href="/facts/Surface_(topology)/jK1b1KIG">surfaces</a> of genus greater than 0 are aspherical (as they have either the Euclidean plane or the hyperbolic plane as a universal cover).</li>
<li>It follows that all non-orientable surfaces, except the real <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a>, are aspherical as well, as they can be covered by an orientable surface of genus 1 or higher.</li>
<li>Similarly, a <a href="/facts/Product_topology/0hg02IaX">product</a> of any number of <a href="/facts/Circle/9fy5ggKn">circles</a> is aspherical. As is any complete, Riemannian flat manifold.</li>
<li>Any <a href="/facts/Hyperbolic_3-manifold/U8AAYhsw">hyperbolic 3-manifold</a> is, by definition, covered by the hyperbolic 3-space H3, hence aspherical. As is any <i>n</i>-manifold whose universal covering space is hyperbolic <i>n</i>-space H<i>n</i>.</li>
<li>Let <i>X</i> = <i>G</i>/<i>K</i> be a <a href="/facts/Riemannian_symmetric_space/AWYuHfNp">Riemannian symmetric space</a> of negative type, and Γ be a <a href="/facts/Lattice_(discrete_subgroup)/fLr9dYhz">lattice</a> in <i>G</i> that acts freely on <i>X</i>. Then the  <a href="/facts/Locally_symmetric_space/AWYuHfNp">locally symmetric space</a> 
  
    
      
        Γ
        ∖
        G
        
          /
        
        K
      
    
    {\displaystyle \Gamma \backslash G/K}
  
 is aspherical.</li>
<li>The <a href="/facts/Bruhat%25E2%2580%2593Tits_building/916RYbEN">Bruhat–Tits building</a> of a simple <a href="/facts/Algebraic_group/DDtBCIvh">algebraic group</a> over a field with a <a href="/facts/Discrete_valuation/Q54hIUDJ">discrete valuation</a> is aspherical.</li>
<li>The complement of a <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a> in S3 is aspherical, by the <a href="/facts/Sphere_theorem_(3-manifolds)/2rwAVmAf">sphere theorem</a></li>
<li>Complete metric spaces with nonpositive curvature in the sense of <a href="/facts/Aleksandr_Aleksandrov_(mathematician)/kBQImYBH">Aleksandr D. Aleksandrov</a> (locally <a href="/facts/CAT(0)_space/fmnJjYsN">CAT(0) spaces</a>) are aspherical. In the case of <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifolds</a>, this follows from the <a href="/facts/Cartan%25E2%2580%2593Hadamard_theorem/Hu5vWBVJ">Cartan–Hadamard theorem</a>, which has been generalized to <a href="/facts/Geodesic_metric_space/e0aAePjd">geodesic metric spaces</a> by <a href="/facts/Mikhail_Gromov_(mathematician)/N0t5hhcI">Mikhail Gromov</a> and <a href="/facts/Hans_Werner_Ballmann/oPC84Efb">Hans Werner Ballmann</a>. This class of aspherical spaces subsumes all the previously given examples.</li>
<li>Any <a href="/facts/Nilmanifold/osYBvj2u">nilmanifold</a> is aspherical.</li></ul>
<h2 id="symplectically-aspherical-manifolds">Symplectically aspherical manifolds</h2>
<p>In the context of <a href="/facts/Symplectic_manifold/JvNrJkC7">symplectic manifolds</a>, 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
</p>

∫
          
            
              S
              
                2
              
            
          
        
        
          f
          
            ∗
          
        
        ω
        =
        ⟨
        
          c
          
            1
          
        
        (
        T
        M
        )
        ,
        
          f
          
            ∗
          
        
        [
        
          S
          
            2
          
        
        ]
        ⟩
        =
        0
      
    
    {\displaystyle \int _{S^{2}}f^{*}\omega =\langle c_{1}(TM),f_{*}[S^{2}]\rangle =0}

<p>for every continuous mapping
</p>

f
        :
        
          S
          
            2
          
        
        →
        M
        ,
      
    
    {\displaystyle f\colon S^{2}\to M,}

<p>where 
  
    
      
        
          c
          
            1
          
        
        (
        T
        M
        )
      
    
    {\displaystyle c_{1}(TM)}
  
 denotes the first <a href="/facts/Chern_class/Tw2WD5hM">Chern class</a> of an <a href="/facts/Almost_complex_manifold/wsXTsuC6">almost complex structure</a> which is compatible with ω.
</p><p>By <a href="/facts/Stokes%2527_theorem/z0pAoj2x">Stokes' theorem</a>, 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.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>Some references<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> drop the requirement on <i>c</i>1 in their definition of "symplectically aspherical."   However, it is more common for symplectic manifolds satisfying only this weaker condition to be called "weakly exact."
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Acyclic_space/IMdGPDGh">Acyclic space</a></li>
<li><a href="/facts/Essential_manifold/PKBNxMGR">Essential manifold</a></li>
<li><a href="/facts/Whitehead_conjecture/5hV6jvmP">Whitehead conjecture</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Martin_Bridson/q63MNtcW">Bridson, Martin R.</a>; <a href="/facts/Andr%25C3%25A9_Haefliger/VFEy0jaS">Haefliger, André</a> (1999). <i>Metric Spaces of Non-Positive Curvature</i>. Grundlehren der mathematischen Wissenschaften. Vol. 319. Berlin, Heidelberg: <a href="/facts/Springer_Science%252BBusiness_Media/nAesf6nT">Springer</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-662-12494-9">10.1007/978-3-662-12494-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-08399-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1744486">1744486</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20110719102802/http://www.map.him.uni-bonn.de/index.php/Aspherical_manifolds">Aspherical manifolds</a> on the Manifold Atlas.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Gompf, Robert E. (1998). "Symplectically aspherical manifolds with nontrivial π2". Mathematical Research Letters. 5 (5): 599–603. arXiv:math/9808063. CiteSeerX 10.1.1.235.9135. doi:10.4310/MRL.1998.v5.n5.a4. MR 1666848. S2CID 15738108. <a href="/wiki/Robert_Gompf" target="_blank">/wiki/Robert_Gompf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kedra, Jarek; Rudyak, Yuli; Tralle, Aleksey (2008). "Symplectically aspherical manifolds". Journal of Fixed Point Theory and Applications. 3: 1–21. arXiv:0709.1799. CiteSeerX 10.1.1.245.455. doi:10.1007/s11784-007-0048-z. MR 2402905. S2CID 13630163. <a href="/wiki/Yuli_Rudyak" target="_blank">/wiki/Yuli_Rudyak</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Aspherical space open-in-new

Aspherical space