# [seqfan] How many manifolds of genus n?

Jonathan Post 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.

http://mathworld.wolfram.com/EulerCharacteristic.html

```