Heegaard Splitting

In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

Definitions

Let ''V'' and ''W'' be handlebodies of genus ''g'', and let ƒ be an orientation reversing homeomorphism from the boundary of ''V'' to the boundary of ''W''. By gluing ''V'' to ''W'' along ƒ we obtain the compact oriented 3-manifold

:$M = V \cup_f W.$

Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.

The decomposition of ''M'' into two handlebodies is called a Heegaard splitting, and their common boundary ''H'' is called the Heegaard surface of the splitting. Splittings are considered up to isotopy.

The gluing map ƒ need only be specified up to taking a double coset in the mapping class group of ''H''. This connection with the mapping class group was first made by W. B. R. Lickorish.

Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies. The gluing map is between the positive boundaries of the compression bodies.

A closed curve is called essential if it is not homotopic to a point, a puncture, or a boundary component.

A Heegaard splitting is reducible if there is an essential simple closed curve $\alpha$ on ''H'' which bounds a disk in both ''V'' and in ''W''. A splitting is irreducible if it is not reducible. It follows from Haken's Lemma that in a reducible manifold every splitting is reducible.

A Heegaard splitting is stabilized if there are essential simple closed curves $\alpha$ and $\beta$ on ''H'' where $\alpha$ bounds a disk in ''V'', $\beta$ bounds a disk in ''W'', and $\alpha$ and $\beta$ intersect exactly once. It follows from Waldhausen's Theorem that every reducible splitting of an irreducible manifold is stabilized.

A Heegaard splitting is weakly reducible if there are disjoint essential simple closed curves $\alpha$ and $\beta$ on ''H'' where $\alpha$ bounds a disk in ''V'' and $\beta$ bounds a disk in ''W''. A splitting is strongly irreducible if it is not weakly reducible.

A Heegaard splitting is minimal or minimal genus if there is no other splitting of the ambient three-manifold of lower genus. The minimal value ''g'' of the splitting surface is the Heegaard genus of ''M''.

Generalized Heegaard splittings

A generalized Heegaard splitting of ''M'' is a decomposition into compression bodies $V_i, W_i, i = 1, \dotsc, n$ and surfaces $H_i, i = 1, \dotsc, n$ such that $\partial_+ V_i = \partial_+ W_i = H_i$ and $\partial_- W_i = \partial_- V_$. The interiors of the compression bodies must be pairwise disjoint and their union must be all of $M$. The surface $H_i$ forms a Heegaard surface for the submanifold $V_i \cup W_i$ of $M$. (Note that here each ''V_i'' and ''W_i'' is allowed to have more than one component.)

A generalized Heegaard splitting is called strongly irreducible if each $V_i \cup W_i$ is strongly irreducible.

There is an analogous notion of thin position, defined for knots, for Heegaard splittings. The complexity of a connected surface ''S'', ''c(S)'', is defined to be $\operatorname\left\$; the complexity of a disconnected surface is the sum of complexities of its components. The complexity of a generalized Heegaard splitting is the multi-set ', where the index runs over the Heegaard surfaces in the generalized splitting. These multi-sets can be well-ordered by lexicographical ordering (monotonically decreasing). A generalized Heegaard splitting is thin if its complexity is minimal.

Examples

; Three-sphere: The three-sphere $S^3$ is the set of vectors in $\mathbb^4$ with length one. Intersecting this with the $xyz$ hyperplane gives a two-sphere. This is the standard genus zero splitting of $S^3$. Conversely, by Alexander's Trick, all manifolds admitting a genus zero splitting are homeomorphic to $S^3$. Under the usual identification of $\mathbb^4$ with $\mathbb^2$ we may view $S^3$ as living in $\mathbb^2$. Then the set of points where each coordinate has norm $1/\sqrt$ forms a Clifford torus, $T^2$. This is the standard genus one splitting of $S^3$. (See also the discussion at Hopf bundle.)
; Stabilization: Given a Heegaard splitting ''H'' in ''M'' the stabilization of ''H'' is formed by taking the connected sum of the pair $(M, H)$ with the pair $\left(S^3, T^2\right)$. It is easy to show that the stabilization procedure yields stabilized splittings. Inductively, a splitting is standard if it is the stabilization of a standard splitting.
; Lens spaces: All have a standard splitting of genus one. This is the image of the Clifford torus in $S^3$ under the quotient map used to define the lens space in question. It follows from the structure of the mapping class group of the two-torus that only lens spaces have splittings of genus one.
; Three-torus: Recall that the three-torus $T^3$ is the Cartesian product of three copies of $S^1$ (circles). Let $x_0$ be a point of $S^1$ and consider the graph $\Gamma = S^1 \times \ \times \ \cup \ \times S^1 \times \ \cup \ \times \ \times S^1$. It is an easy exercise to show that ''V'', a regular neighborhood of $\Gamma$, is a handlebody as is $T^3 - V$. Thus the boundary of ''V'' in $T^3$ is a Heegaard splitting and this is the standard splitting of $T^3$. It was proved by Charles Frohman and Joel Hass

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