Aspherical space explained
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).
Examples
- Using the second of above definitions we easily see that all orientable compact surfaces 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, are aspherical as well, as they can be covered by an orientable surface of genus 1 or higher.
- Similarly, a product of any number of circles is aspherical. As is any complete, Riemannian flat manifold.
- Any hyperbolic 3-manifold is, by definition, covered by the hyperbolic 3-space H3, hence aspherical. As is any n-manifold whose universal covering space is hyperbolic n-space Hn.
is aspherical.
Symplectically aspherical manifolds
In the context of symplectic manifolds, 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
for every continuous mapping
where
denotes the first
Chern class of an
almost complex structure which is compatible with ω.
By 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.[1]
Some references[2] drop the requirement on c1 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
Notes
- Robert E. . Gompf . Robert Gompf. Symplectically aspherical manifolds with nontrivial π2 . Mathematical Research Letters . 5 . 5 . 599–603 . 1998 . 10.4310/MRL.1998.v5.n5.a4 . 1666848 . math/9808063 . 10.1.1.235.9135. 15738108 .
- Jarek . Kedra . Yuli . Rudyak . Yuli Rudyak . Aleksey . Tralle . Symplectically aspherical manifolds . Journal of Fixed Point Theory and Applications . 3 . 1–21 . 2008 . 10.1007/s11784-007-0048-z . 2402905 . 0709.1799 . 10.1.1.245.455. 13630163 .
References
- Book: Bridson. Martin R.. Martin Bridson. Metric Spaces of Non-Positive Curvature. Haefliger. André. André Haefliger. 1999. Springer. 978-3-642-08399-0. Grundlehren der mathematischen Wissenschaften. 319. Berlin, Heidelberg. 10.1007/978-3-662-12494-9. 1744486.
External links