Handle Decomposions for $2$ Manifolds

general-topologysurgery-theory

I have a question about the notation for "$n$-handles" with respect to a decomposion of handlebodies.

At german wikipedia page https://de.wikipedia.org/wiki/Henkel-Zerlegung

I found a statement that every closed orientable $2$-manifold has following handle decomposion:

-one $0$-handle (so a punctured $2$-sphere)

–$2g$ $1$-handles (cylinders) where $g$ is the genus of the surface

-one $2$-handle

Could anybody explain/ visualize to me this decomposion?

Best Answer

When constructing an $n$-manifold, a $j$-handle is an $B^{j} \times B^{n-j}$, where $B^{k}$ is a $k$-ball and is attached to the rest of the manifold along $S^{j-1} \times B^{n-j}$, where $S^k$ is a $k$-sphere.

Since you are making a $2$-manifold,

  • A $0$-handle is a $B^0 \times B^2$ (a disk) glued along $S^{-1} \times B^2 = \varnothing$. (Since we start with a $0$-handle, there is nothing to glue it to.)
  • A $1$-handle is a $B^1 \times B^1$ (a rectangular strip) glued along $S^0 \times B^1$ (two opposite ends of the strip).
  • A $2$-handle is a $B^2 \times B^0$ (homeomorphic to a disk) glued along $S^1 \times B^0$ (homeomorphic to a circle).

This image (from "Efficient Edgebreaker for surfaces of arbitrary topology" by Lewiner, Lopes, Rossignac, Vieira) starts with a $0$-handle in the upper-left, attaches a $1$-handle, then another, and finally fills in the "gap" between the unglued sides of the $1$-handles with a $2$-handle. (Montesinos, Classical Tesselations and Three-Manifolds, discusses this, with diagrams, in section 1.2.)

An equivalent way to specify a $2$-manifold is by identifications of a disk. Examples. The equivalence of these two descriptions is fairly straightforward.

  • The edges of the polygon are given orientations and labels, each label appearing on two edges. The polygon is the $0$-handle.
  • Each pair of edges having the same label are attached by a $1$-handle, respecting the orientation of the edges.
  • Finally a $2$-handle is glued along the (long) circle that is the union of the unglued boundaries of the $1$-handles. (When thinking about the polygon with identifications, this $2$-handle is the identification of all the vertices of the polygon to one point.)

Let $M$ be the oriented, closed, genus $g$ $2$-manifold. It has a specification by a polygon with $2g$ edges. To specify the labels and orientations, we pick an orientation for the $2$-cell filling the polygon, which induces an orientation on the boundary (either clockwise or anticlockwise). Then we indicate by a superscript on the label whether the edge is oriented in the same direction ("$1$") or in the opposite direction ("$-1$"). Then a labelling of the polygon with identifications producing $M$ is $a_1 b_1 a_1^{-1} b_1^{-1} a_2 b_2 a_2^{-1} b_2^{-1} \dots a_g b_g a_g^{-1} b_g^{-1}$. As a handle decomposition:

  • Start with the polygon (the $0$-handle) and pick an edge and a direction to proceed. WLOG, I pick clockwise to simplify the following description.
  • Attach a $1$-handle to the edge and to the edge two edges clockwise, without any twists on the $1$-handle. Advance clockwise to the next edge.
  • Attach a $1$-handle to the edge and to the edge two edges clockwise, without any twists on the $1$-handle. Advance clockwise by three edges (since the next two already hanve handles attached).
  • Continue attaching pairs of handles as described above until all $2g$ edges have a handle attached. Then attach a $2$-handle to the circle formed by the unglued edges of all the $1$-handles.
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