[seqfan] How many manifolds of genus n?
jvospost3 at gmail.com
Sun May 27 15:41:34 CEST 2012
How many closed manifolds of genus n?
The unique closedmanifold of genus 0 is the sphere, so qa(0) = 1.
For genus 1, we have two: torus, and projective plane, so a(1) = 2.
For genus 2 we have at least two; Klein bottle, and 2-handled sphere,
so a(2) => 2.
Genus is topologically invariant property of a surface defined as the
largest number of nonintersecting simple closed curves that can be
drawn on the surface without separating it. Roughly speaking, it is
the number of holes in a surface.
The genus of a surface, also called the geometric genus, is related to
the Euler characteristic Chi. For a orientable surface such as a
sphere (genus 0) or torus (genus 1), the relationship is
Chi = 2 - 2g
For a nonorientable surface such as a real projective plane (genus 1)
or Klein bottle (genus 2), the relationship is
Chi = 2 - g
Let a closed surface have genus g. Then the polyhedral formula
generalizes to the Poincaré formula
Chi(g) = V - E + F
I limit the question to closed manifolds, excluding, for instance,
cylinder and mobius strip (Chi = 0)
We know that every compact 3-manifold is the connected sum of a unique
collection of prime 3-manifolds.
We know more, not immediately useful to our question. For example,
Jaco-Shalen-Johannson Torus Decomposition:
Irreducible orientable compact 3-manifolds have a canonical (up to
isotopy) minimal collection of disjointly embedded incompressible tori
such that each component of the 3-manifold removed by the tori is
either "atoroidal" or "Seifert-fibered."
Cf. A156097 Number of rigid genus-2 bipartite crystallizations of
orientable 3-manifolds with 2n vertices.
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