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I was looking through a few papers in Applied math and I saw something strange:

First, in this paper

"...the dimension of Teichmüller space of genus 0 surfaces with $n$ boundaries is $3n-6$."

And also in this paper

"The dimension of the Teichmüller space of negative Euler number surfaces with topological type (g>1,r) is $6g−5+3r$... The teapot surface has one handle and one boundary at the spout, therefore it is of topological type (1,1), with 3 dimensions in Teichmüller space."

From what I know about Teichmüller spaces (of hyperbolic Riemann Surfaces), their dimensions are given by

  1. $6g-6+2n$ for genus $g$ surfaces with $n$ missing points.
  2. Infinite dimension if they are missing a disk.

So I am very confused about these numbers, especially the $6g-5+3r$, since $T(X)$ must be even dimensional.

I feel like I am missing something basic?

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  • $\begingroup$ I think $6g - 6 + 2n$ only holds with hypotheses on $(g, n)$, e.g. it clearly doesn't hold for $(g, n) = (0, 0), (1, 0)$. $\endgroup$ – Qiaochu Yuan Feb 2 '15 at 7:12
  • $\begingroup$ @QiaochuYuan Yes, the dimension $6g-6+2n$ is for those with negative Euler characteristic. $\endgroup$ – Braindead Feb 2 '15 at 12:33
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There's a distinction between punctures and boundaries. Punctures are what give the $6g-6+2n$ formula you know. (Intuitively, the $2$ for each puncture is "two degrees of freedom for where to place each puncture.")

Boundaries are different. By a boundary here, I mean a geodesic boundary, necessarily with positive length. So there's the length parameter adding the extra degree of freedom, making this "three degrees of freedom per boundary," intuitively speaking.

More rigorously, in terms of the Fenchel-Nielsen coordinates for the case with boundary: Chop up your surface into pairs of pants with geodesic boundary (in particular, we have each boundary curve as a boundary of some pair of pants, as well as various internal geodesics to cut along to make pairs of pants). The number of pairs of pants equals the Euler characteristic times $-1$. Suppose we have $b$ boundary components and genus $g$. Then $-1$ times the Euler characteristic is $2g-2 + b$. Now for each pair of pants we have 3 length parameters, giving $6g-6+3b$ total length parameters. We have $6g-6+3b$ total cuffs (after cutting the surface apart into disjoint pants). There are $b$ boundary cuffs that don't need to be glued to another cuff, leaving $3g-3+b$ gluings of a pair of cuffs. That gives $3g-3+b$ additional twist parameters, but we must match up lengths to glue, so we lose $3g-3+b$ length parameters as well. This leaves $6g-6+3b$ as the dimension.

Notice for genus zero this is $3b-6$ as you have above.

I suspect the $6g-5+3r$ is a typo and should be $6g-6+3r$ (where $r$ is the number of boundaries). Indeed, for the example $(g,r) = (1,1)$ you quote $3$ as the dimension, which agrees with $6g-6+3r$. (Alternatively, perhaps one of the $r$ "boundaries" [but only one] is supposed to be a puncture? I didn't read the paper.)

The formula in general is $6g-6+2p+3b$ where $p$ is the number of punctures and $b$ is the number of boundary components.


Definitions (complex structures versus hyperbolic metrics):

There are two traditional definitions of Teichmüller space for a (smooth, just to fix conventions) compact surface $\Sigma$: 1) The space of complex structures on $\Sigma$ modulo identifications by diffeomorphisms isotopic to the identity. 2) The space of metrics on $\Sigma$ of constant curvature $-1$ modulo identifications by diffeomorphisms isotopic to the identity.

The former is naturally a complex manifold. The latter a priori only has the structure of a real manifold, and is the underlying real manifold for the former.

In case you haven't seen this before, the correspondence between the two: A complex structure determines a conformal class of metrics on the real $2$-manifold $\Sigma$; that is, a metric up to multiplication by a function from $\Sigma \rightarrow \mathbb{R}$. Namely, the class of metrics for which $v$ and $iv$ (for now we have a complex structure on $T\Sigma$) are orthogonal and of the same length for any tangent vector $v$. Then, given some metric $g$ in this conformal class, it turns out there exists a unique solution $f: \Sigma \rightarrow \mathbb{R}$ to the PDE given by "the curvature of $fg$ is constant and equal to $-1$" (this requires some analysis background to prove).

It turns out this extends to the case with punctures if we require the hyperbolic metric to be complete (for then this is no longer guaranteed by compactness).

For the case with boundary, for hyperbolic manifolds one requires that the boundary is a geodesic curve. As you mention, the space would be infinite dimensional otherwise. The space of these is the fixed point set of an involution on the Teichmüller space of the "doubled" surface (i.e. the surface you get by gluing two copies of the original surface together at corresponding boundary components). On the complex side, this is also the fixed point set of an involution, but this fixed point set is no longer a complex manifold, and is rather a real manifold of half the dimension (much like $\mathbb{R} \subset \mathbb{C}$ is the fixed point set of complex conjugation).

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  • $\begingroup$ I feel like the definition of Teichmuller space you are describing is not the same as the one I am accustomed to. The definition I am familiar with makes $T(X)$ a complex manifold, which forces $T(X)$ to have even dimensions. So it cannot have a dimension of 3. $\endgroup$ – Braindead Mar 8 '15 at 19:24
  • $\begingroup$ Like I stated in my post, the definition I am used to forces $T(X)$ to be infinite dimensional when there are boundaries. (en.wikipedia.org/wiki/Teichm%C3%BCller_space#Properties_of_TX) $\endgroup$ – Braindead Mar 8 '15 at 19:27
  • $\begingroup$ @Braindead I've added a section discussing different types of Teichmüller spaces and the relationships between them. $\endgroup$ – aes Mar 8 '15 at 23:27

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