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In the mathematical field of geometric topology, a Heegaard splitting Template:IPAc-en 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 simple closed curve on a surface is essential if it does not bound a disc on the surface. This is equivalent to saying that the curve is homologically non-trivial, i.e., the curve is a cycle that does not bound.

A Heegaard splitting is reducible if there is an essential simple closed curve $α$ 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 $α$ and $β$ on H where $α$ bounds a disk in V, $β$ bounds a disk in W, and $α$ and $β$ 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 $α$ and $β$ on H where $α$ bounds a disk in V and $β$ 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 $V_{i}, W_{i}, i = 1, \dots, n$ and surfaces $H_{i}, i = 1, \dots, n$ such that $\partial_{+} V_{i} = \partial_{+} W_{i} = H_{i}$ and $\partial_{-} W_{i} = \partial_{-} V_{i + 1}$ . 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 $\max {0, 1 - χ (S)}$ ; 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 {c(S_i)}, 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 $ℝ^{4}$ with length one. Intersecting this with the $x y z$ 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 $ℝ^{4}$ with $ℂ^{2}$ we may view $S^{3}$ as living in $ℂ^{2}$ . Then the set of points where each coordinate has norm $1 / \sqrt{2}$ 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 $(S^{3}, T^{2})$ . 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 $Γ = S^{1} \times {x_{0}} \times {x_{0}} \cup {x_{0}} \times S^{1} \times {x_{0}} \cup {x_{0}} \times {x_{0}} \times S^{1}$ . It is an easy exercise to show that V, a regular neighborhood of $Γ$ , 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 Frohman and Hass that any other genus 3 Heegaard splitting of the three-torus is topologically equivalent to this one. Boileau and Otal proved that in general any Heegaard splitting of the three-torus is equivalent to the result of stabilizing this example.

Theorems

Alexander's Lemma: Up to isotopy, there is a unique (piecewise linear) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the Schoenflies theorem. In dimension two this is the Jordan curve theorem.) This may be restated as follows: the genus zero splitting of $S^{3}$ is unique.

Waldhausen's Theorem: Every splitting of $S^{3}$ is obtained by stabilizing the unique splitting of genus zero.

Suppose now that M is a closed orientable three-manifold.

Reidemeister-Singer Theorem: For any pair of splittings $H_{1}$ and $H_{2}$ in M there is a third splitting $H$ in M which is a stabilization of both.

Haken's Lemma: Suppose that $S_{1}$ is an essential two-sphere in M and H is a Heegaard splitting. Then there is an essential two-sphere $S_{2}$ in M meeting H in a single curve.

Classifications

There are several classes of three-manifolds where the set of Heegaard splittings is completely known. For example, Waldhausen's Theorem shows that all splittings of $S^{3}$ are standard. The same holds for lens spaces (as proved by Francis Bonahon and J.P. Otal).

Splittings of Seifert fiber spaces are more subtle. Here, all splittings may be isotoped to be vertical or horizontal (as proved by Yoav Moriah and Jennifer Schultens).

Template:Harvtxt classified splittings of torus bundles (which includes all three-manifolds with Sol geometry). It follows from their work that all torus bundles have a unique splitting of minimal genus. All other splittings of the torus bundle are stabilizations of the minimal genus one.

As of 2008, the only hyperbolic three-manifolds whose Heegaard splittings are classified are two-bridge knot complements, in a paper of Tsuyoshi Kobayashi.

Applications and connections

Minimal surfaces

Heegaard splittings appeared in the theory of minimal surfaces first in the work of Blaine Lawson who proved that embedded minimal surfaces in compact manifolds of positive sectional curvature are Heegaard splittings. This result was extended by William Meeks to flat manifolds, except he proves that an embedded minimal surface in a flat three-manifold is either a Heegaard surface or totally geodesic.

Meeks and S. T. Yau went on to use results of Waldhausen to prove results about the topological uniqueness of minimal surface of finite topology in $R^{3}$ . The final topological classification of embedded minimal surfaces in $R^{3}$ was given by Meeks and Frohman. The result relied heavily on techniques developed for studying the topology of Heegaard splittings.

Heegaard Floer homology

Heegaard diagrams, which are simple combinatorial descriptions of Heegaard splittings, have been used extensively to construct invariants of three-manifolds. The most recent example of this is the Heegaard Floer homology of Peter Ozsvath and Zoltán Szabó. The theory uses the $g^{t h}$ symmetric product of a Heegaard surface as the ambient space, and tori built from the boundaries of meridian disks for the two handlebodies as the Lagrangian submanifolds.

History

The idea of a Heegaard splitting was introduced by Template:Harvs. While Heegaard splittings were studied extensively by mathematicians such as Wolfgang Haken and Friedhelm Waldhausen in the 1960s, it was not until a few decades later that the field was rejuvenated by Template:Harvtxt, primarily through their concept of strong irreducibility.

References

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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Contents

Definitions

Generalized Heegaard splittings

Examples

Theorems

Classifications

Applications and connections

Minimal surfaces

Heegaard Floer homology

History

See also

References

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Light field

Definitions

Generalized Heegaard splittings

Examples

Theorems

Classifications

Applications and connections

Minimal surfaces

Heegaard Floer homology

History

See also

References

Navigation menu

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