Hilbert's inequality: Difference between revisions
en>Makecat m clean up, typos fixed: it it → it is, is is → is using AWB (8062) |
en>FrescoBot m Bot: fixing section wikilinks and minor changes |
||
Line 1: | Line 1: | ||
{{Orphan|date=May 2013}} | |||
In [[mathematics]], a '''quasitoric manifold''' is a topological analogue of the [[Toric_variety#Resolution of Singularities|nonsingular]] projective [[toric variety]] of [[algebraic geometry]]. A [[Smooth manifold|smooth]] <math>2n</math>-dimensional [[manifold]] is a quasitoric manifold if it admits a smooth, locally standard action of an [[Torus#n-dimensional_torus|<math>n</math>-dimensional torus]], with [[orbit space]] an <math>n</math>-dimensional [[Simple polytope|simple]] [[Convex polytope|convex]] [[polytope]]. | |||
Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz,<ref name=autogenerated1>M. Davis and T. Januskiewicz, 1991.</ref> who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth [[toric variety|toric varieties]], which are known to algebraic geometers as [[toric manifold]]s.<ref>V. Buchstaber and T. Panov, 2002.</ref> | |||
Quasitoric manifolds are studied in a variety of contexts in [[algebraic topology]], such as [[Complex cobordism|complex cobordism theory]], and the other oriented [[Cohomology theory|cohomology theories]].<ref>V. Buchstaber and N. Ray, 2008.</ref> | |||
==Definitions== | |||
Denote the <math>i</math>-th subcircle of the <math>n</math>-torus <math>T^n</math> by <math>T_i</math> so that <math>T_1 \times \ldots \times T_n = T^n</math>. Then coordinate-wise multiplication of <math>T^n</math> on <math>\mathbb{C}^n</math> is called the '''standard representation'''. | |||
Given open sets <math>X</math> in <math>M^{2n}</math> and <math>Y</math> in <math>\mathbb{C}^n</math>, that are closed under the [[Group action|action]] of <math>T^n</math>, a <math>T^{n}</math>-action on <math>M^{2n}</math> is defined to be '''locally isomorphic''' to the standard representation if <math>h(tx) = \alpha(t)h(x)</math>, for all <math>t</math> in <math>T^n</math>, <math>x</math> in <math>X</math>, where <math>h</math> is a [[homeomorphism]] <math>X \rightarrow Y</math>, and <math>\alpha</math> is an [[automorphism]] of <math>T^n</math>. | |||
Given a simple convex polytope <math>P^n</math> with <math>m</math> [[Facet (mathematics)|facets]], a <math>T^n</math>-manifold <math>M^{2n}</math> is a '''quasitoric manifold over''' <math>P^n</math> if, | |||
# the <math>T^n</math>-action is locally isomorphic to the standard representation, | |||
# there is a [[projection map|projection]] <math>\pi : M^{2n} \rightarrow P^n</math> that maps each <math>l</math>-dimensional orbit to a point in the interior of an <math>l</math>-dimensional [[Face (geometry)|face]] of <math>P^n</math>, for <math>l = 0,</math> <math>...,</math> <math>n</math>. | |||
The definition implies that the fixed points of <math>M^{2n}</math> under the <math>T^n</math>-action are mapped to the vertices of <math>P^n</math> by <math>\pi</math>, while points where the action is free project to the interior of the polytope. | |||
==The dicharacteristic function== | |||
A quasitoric manifold can be described in terms of a '''dicharacteristic function''' and an associated '''dicharacteristic matrix'''. In this setting it is useful to assume that the facets <math>F_1,\dots,F_m</math> of <math>P^n</math> are ordered so that the intersection <math>F_{1} \cap \dots \cap F_{n}</math> is a vertex <math>v</math> of <math>P^{n}</math>, called the '''initial vertex'''. | |||
A '''dicharacteristic function''' is a [[homomorphism]] <math>\lambda : T^m \rightarrow T^n</math>, such that if <math>F_{i_1} \cap \dots \cap F_{i_k}</math> is a [[codimension]]-<math>k</math> face of <math>P^n</math>, then <math>\lambda</math> is a [[monomorphism]] on restriction to the subtorus <math>T_{i_1} \times \dots \times T_{i_k}</math> in <math>T^m</math>. | |||
The restriction of λ to the subtorus <math>T_1 \times \ldots \times T_n</math> corresponding to the initial vertex <math>v</math> is an isomorphism, and so <math>\lambda(T_1), \ldots, \lambda(T_n)</math> can be taken to be a basis for the [[Lie algebra]] of <math>T^n</math>. The [[epimorphism]] of Lie algebras associated to λ may be described as a linear transformation <math>\mathbb{Z}^m \rightarrow \mathbb{Z}^n</math>, represented by the <math> n \times m</math> '''dicharacteristic matrix''' <math>\Lambda</math> given by | |||
::<math> | |||
\begin{bmatrix} | |||
1 & 0 & \dots & 0 & \lambda_{1, n+1} & \dots & \lambda_{1, m}\\ | |||
0 & 1 & \dots & 0 & \lambda_{2, n+1} & \dots & \lambda_{2, m}\\ | |||
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ | |||
0 & 0 & \dots & 1 & \lambda_{n, n+1} & \dots & \lambda_{n, m} | |||
\end{bmatrix}. | |||
</math> | |||
The <math>i</math>th column of <math>\Lambda</math> is a primitive vector <math>\lambda_i = (\lambda_{1, i}, \dots, \lambda_{n, i})</math> in <math>\mathbb{Z}^n</math>, called the '''facet vector'''. As each facet vector is primitive, whenever the facets <math>F_{i_1} \cap \dots \cap F_{i_n}</math> meet in a vertex, the corresponding columns <math>\lambda_{i_1}, \dots \lambda_{i_n}</math> form a basis of <math>\mathbb{Z}^n</math>, with determinant equal to <math>\pm 1</math>. The [[Isotropy_group#Orbits_and_stabilizers|isotropy subgroup]] associated to each facet <math>F_i</math> is described by | |||
::<math> \{(e^{2\pi i \theta\lambda_{1,i}}, \ldots, e^{2\pi i \theta\lambda_{n,i}}) \in T^n \}, </math> | |||
for some <math>\theta</math> in <math>\mathbb{R}</math>. | |||
In their original treatment of quasitoric manifolds, Davis and Januskiewicz<ref name=autogenerated1 /> introduced the notion of a '''characteristic function''' that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle <math>\lambda(T_i)</math> be oriented, forcing a choice of sign for each vector <math>\lambda_i</math>. The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray<ref>V. Buchstaber and N. Ray, 2001.</ref> to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix <math>\Lambda</math> as <math>(I_n \mid S)</math>, where <math>I_n</math> is the [[identity matrix]] and <math>S</math> is an <math>n \times (m-n)</math> submatrix.<ref>V. Buchstaber, T. Panov and N. Ray, 2007.</ref> | |||
==Relation to the moment-angle complex== | |||
The kernel <math>K(\lambda)</math> of the dicharacteristic function acts freely on the [[moment angle complex]] <math>Z_{P^n}</math>, and so defines a [[Principle fiber bundle|principal]] <math>K(\lambda)</math>-bundle <math>Z_{P^n} \rightarrow M^{2n}</math> over the resulting [[quotient space]] <math>M^{2n}</math>. This quotient space can be viewed as | |||
:: <math>T^n \times P^n / \sim, \, </math> | |||
where pairs <math>(t_1, p_1)</math>, <math>(t_2, p_2)</math> of <math>T^n \times P^n</math> are identified if and only if <math>p_1 = p_2</math> and <math>t_1^{-1}t_2</math> is in the image of <math>\lambda</math> on restriction to the subtorus <math>T_{i_1} \times \dots \times T_{i_k}</math> that corresponds to the unique face <math>F_{i_1} \cap \dots \cap F_{i_k}</math> of <math>P^n</math> containing the point <math>p_1</math>, for some <math>1 \leq k \leq n</math>. | |||
It can be shown that any quasitoric manifold <math>M^{2n}</math> over <math>P^n</math> is [[Equivariant map|equivariently]] [[Diffeomorphism|diffeomorphic]] to a quasitoric manifold of the form of the quotient space above.<ref>M. Davis and T. Januskiewicz, 1991, Proposition 1.8.</ref> | |||
==Examples== | |||
* The <math>n</math>-dimensional [[complex projective space]] <math>\mathbb{C}P^n</math> is a quasitoric manifold over the <math>n</math>-[[simplex]] <math>\Delta^n</math>. If <math>\Delta^n</math> is embedded in <math>\mathbb{R}^{n+1}</math> so that the origin is the initial vertex, a dicharacteristic function can be chosen so that the associated dicharacteristic matrix is | |||
::<math> | |||
\begin{bmatrix} | |||
1 & 0 & \dots & 0 & -1\\ | |||
0 & 1 & \dots & 0 & -1\\ | |||
\vdots & \vdots & \ddots & \vdots & \vdots \\ | |||
0 & 0 & \dots & 1 & -1 | |||
\end{bmatrix}. | |||
</math> | |||
The moment angle complex <math>Z_{\Delta^n}</math> is the <math>(2n+1)</math>-sphere <math>S^{2n+1}</math>, the kernel <math>K(\lambda)</math> is the diagonal subgroup <math>\{(t, \dots, t)\} < T^{n+1}</math>, so the quotient of <math>Z_{\Delta^n}</math> under the action of <math>K(\lambda)</math> is <math>\mathbb{C}P^n</math>.<ref>V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.11.</ref> | |||
* The [[Bott manifold]]s that form the stages in a [[Bott tower]] are quasitoric manifolds over [[Hypercube|<math>n</math>-cubes]]. The <math>n</math>-cube <math>I^n</math> is embedded in <math>\mathbb{R}^{2n}</math> so that the origin is the initial vertex, and a dicharacteristic function is chosen so that the associated dicharacteristic matrix <math>(I_n \mid S)</math> has <math>S</math> given by | |||
::<math> | |||
\begin{bmatrix} | |||
1 &0 &\cdots &0 &0 &\cdots &0 &0 \\ | |||
-a(1,2) &1 &\cdots &0 &0 &\cdots &0 &0 \\ | |||
\vdots &\vdots & &\vdots &\vdots & &\vdots &\vdots \\ | |||
-a(1,i) &-a(2,i) &\cdots &-a(i-1,i) &1 &\cdots &0 &0 \\ | |||
\vdots &\vdots & &\vdots &\vdots & &\vdots &\vdots \\ | |||
-a(1,n) &-a(2,n) &\cdots &-a(i-1,n) &-a(i,n) &\cdots &-a(n-1,n) &1 | |||
\end{bmatrix}, | |||
</math> | |||
for integers <math>a(i,j)</math>. | |||
The moment angle complex <math>Z_{I^n}</math> is a product of <math>n</math> copies of 3-sphere embedded in <math>\mathbb{C}^{2n}</math>, the kernel <math>K(\lambda)</math> is given by | |||
::<math>\{(t_1,t_1^{-a(1,2)}t_2,\dots,t_1^{-a(1,i)}\dots t_{i-1}^{-a(i-1,i)}t_i,\dots, t_1^{-a(1,n)}\dots t_{n-1}^{-a(n-1,n)}t_n, t_1^{-1}, \dots, t_n^{-1}) : t_i \in T, 1\leq i\leq n \} < T^{2n}</math>, | |||
so that the quotient of <math>Z_{I^n}</math> under the action of <math>K(\lambda)</math> is the <math>n</math>-th stage of a Bott tower.<ref>V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.12.</ref> The integer values <math>a(i,j)</math> are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower.<ref>Y. Civan and N. Ray, 2005.</ref> | |||
==The cohomology ring of a quasitoric manifold== | |||
Canonical [[complex line bundle]]s <math>\rho_i</math> over <math>M^{2n}</math> given by | |||
:: <math>Z_{P^n} \times_{K(l)} \mathbb{C}_i \longrightarrow M^{2n}</math>, | |||
can be associated with each facet <math>F_i</math> of <math>P^n</math>, for <math>1 \leq i \leq m</math>, where <math>K(\lambda)</math> acts on <math>\mathbb{C}_i</math>, by the restriction of <math>K(\lambda)</math> to the <math>i</math>-th subcircle of <math>T^m</math> embedded in <math>\mathbb{C}</math>. These bundles are known as the '''facial bundles''' associated to the quasitoric manifold. By the definition of <math>M^{2n}</math>, the preimage of a facet <math>\pi^{-1}(F_i)</math> is a <math>2(n-1)</math>-dimensional quasitoric '''facial submanifold''' <math>M_i</math> over <math>F_i</math>, whose isotropy subgroup is the restriction of <math>\lambda</math> on the subcircle <math>T_i</math> of <math>T^m</math>. Restriction of <math>\rho_i</math> to <math>M_i</math> gives the [[Normal bundle|normal]] 2-plane bundle of the embedding of <math>M_i</math> in <math>M^{2n}</math>. | |||
Let <math>x_i</math> in <math>H^2(M^{2n}; \mathbb{Z})</math> denote the first [[Chern class]] of <math>\rho_i</math>. The integral [[cohomology ring]] <math>H^*(M^{2n}; \mathbb{Z})</math> is generated by <math>x_i</math>, for <math>1 \leq i \leq m</math>, subject to two sets of relations. The first are the relations generated by the [[Stanley–Reisner ring|Stanley–Reisner ideal]] of <math>P^n</math>; linear relations determined by the dicharacterstic function comprise the second set: | |||
:: <math>x_i = -\lambda_{i, n+1}x_{n+1} - \cdots - \lambda_{i, m}x_m, \mbox{ for } 1\leq i \leq n</math>. | |||
Therefore only <math>x_{n+1}, \dots, x_m</math> are required to generate <math>H^*(M^{2n}; \mathbb{Z})</math> multiplicatively.<ref name=autogenerated1 /> | |||
==Comparison with toric manifolds== | |||
* Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds. | |||
* Not all quasitoric manifolds are toric manifolds. For example, the [[connected sum]] <math>\mathbb{C}P^2 \sharp \mathbb{C}P^2</math> can be constructed as a quasitoric manifold, but it is not a toric manifold.<ref>M. Masuda and D. Y. Suh 2007.</ref> | |||
== Notes == | |||
{{Reflist}} | |||
==References== | |||
*{{Citation | last1=Buchstaber | first1=V. | last2=Panov | first2=T. | title=Torus Actions and their Applications in Topology and Combinatorics|year=2002 | publisher=American Mathematical Society | series=University Lecture Series | volume=24}} | |||
*{{Citation | last1=Buchstaber | first1=V. | last2=Panov | first2=T. | last3=Ray | first3=N. | title=Spaces of polytopes and cobordism of quasitoric manifolds |year=2007 | journal=Moscow Mathematical Journal| volume=7 | issue=2 |pages=219–242}} | |||
*{{Citation | last1=Buchstaber | first1=V. | last2=Ray | first2=N. | title=Tangential structures on toric manifolds and connected sums of polytopes |year=2001 | journal=International Mathematics Research Notices | volume=4 | pages=193–219}} | |||
*{{Citation | last1=Buchstaber | first1=V. | last2=Ray | first2=N. | title=Proceedings of the International Conference in Toric Topology; Osaka City University 2006 | year=2008 | publisher=American Mathematical Society| series=Contemporary Mathematics | volume=460 | chapter=An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron | pages=1–27}} | |||
*{{Citation | last1=Civan| first1=Y. | last2=Ray | first2=N. | title=Homotopy decompositions and ''K''-theory of Bott towers|year=2005 | journal=K-Theory | volume=34 | pages=1–33}} | |||
*{{Citation | last1=Davis | first1=M. | last2=Januskiewicz | first2=T. | title=Convex polytopes, Coxeter orbifolds and torus actions |year=1991 | journal=Duke Mathematical Journal | volume=62 | issue=2 | pages=417–451}} | |||
*{{Citation | last1=Masuda | first1=M. | last2=Suh | first2=D. Y. | title=Proceedings of the International Conference in Toric Topology; Osaka City University 2006 | year=2008 | publisher=American Mathematical Society| series=Contemporary Mathematics | volume=460 | chapter=Classification problems of toric manifolds via topology | pages=273–286}} | |||
{{Mathematics-footer}} | |||
{{DEFAULTSORT:Algebraic Topology}} | |||
[[Category:Topology]] | |||
[[Category:Algebraic topology]] | |||
[[Category:Manifolds]] |
Latest revision as of 20:26, 26 January 2013
In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an -dimensional torus, with orbit space an -dimensional simple convex polytope.
Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz,[1] who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties, which are known to algebraic geometers as toric manifolds.[2]
Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.[3]
Definitions
Denote the -th subcircle of the -torus by so that . Then coordinate-wise multiplication of on is called the standard representation.
Given open sets in and in , that are closed under the action of , a -action on is defined to be locally isomorphic to the standard representation if , for all in , in , where is a homeomorphism , and is an automorphism of .
Given a simple convex polytope with facets, a -manifold is a quasitoric manifold over if,
- the -action is locally isomorphic to the standard representation,
- there is a projection that maps each -dimensional orbit to a point in the interior of an -dimensional face of , for .
The definition implies that the fixed points of under the -action are mapped to the vertices of by , while points where the action is free project to the interior of the polytope.
The dicharacteristic function
A quasitoric manifold can be described in terms of a dicharacteristic function and an associated dicharacteristic matrix. In this setting it is useful to assume that the facets of are ordered so that the intersection is a vertex of , called the initial vertex.
A dicharacteristic function is a homomorphism , such that if is a codimension- face of , then is a monomorphism on restriction to the subtorus in .
The restriction of λ to the subtorus corresponding to the initial vertex is an isomorphism, and so can be taken to be a basis for the Lie algebra of . The epimorphism of Lie algebras associated to λ may be described as a linear transformation , represented by the dicharacteristic matrix given by
The th column of is a primitive vector in , called the facet vector. As each facet vector is primitive, whenever the facets meet in a vertex, the corresponding columns form a basis of , with determinant equal to . The isotropy subgroup associated to each facet is described by
In their original treatment of quasitoric manifolds, Davis and Januskiewicz[1] introduced the notion of a characteristic function that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle be oriented, forcing a choice of sign for each vector . The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray[4] to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix as , where is the identity matrix and is an submatrix.[5]
Relation to the moment-angle complex
The kernel of the dicharacteristic function acts freely on the moment angle complex , and so defines a principal -bundle over the resulting quotient space . This quotient space can be viewed as
where pairs , of are identified if and only if and is in the image of on restriction to the subtorus that corresponds to the unique face of containing the point , for some .
It can be shown that any quasitoric manifold over is equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above.[6]
Examples
- The -dimensional complex projective space is a quasitoric manifold over the -simplex . If is embedded in so that the origin is the initial vertex, a dicharacteristic function can be chosen so that the associated dicharacteristic matrix is
The moment angle complex is the -sphere , the kernel is the diagonal subgroup , so the quotient of under the action of is .[7]
- The Bott manifolds that form the stages in a Bott tower are quasitoric manifolds over -cubes. The -cube is embedded in so that the origin is the initial vertex, and a dicharacteristic function is chosen so that the associated dicharacteristic matrix has given by
The moment angle complex is a product of copies of 3-sphere embedded in , the kernel is given by
so that the quotient of under the action of is the -th stage of a Bott tower.[8] The integer values are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower.[9]
The cohomology ring of a quasitoric manifold
Canonical complex line bundles over given by
can be associated with each facet of , for , where acts on , by the restriction of to the -th subcircle of embedded in . These bundles are known as the facial bundles associated to the quasitoric manifold. By the definition of , the preimage of a facet is a -dimensional quasitoric facial submanifold over , whose isotropy subgroup is the restriction of on the subcircle of . Restriction of to gives the normal 2-plane bundle of the embedding of in .
Let in denote the first Chern class of . The integral cohomology ring is generated by , for , subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of ; linear relations determined by the dicharacterstic function comprise the second set:
Therefore only are required to generate multiplicatively.[1]
Comparison with toric manifolds
- Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds.
- Not all quasitoric manifolds are toric manifolds. For example, the connected sum can be constructed as a quasitoric manifold, but it is not a toric manifold.[10]
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- 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 - 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 - 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 - 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
47 year-old Environmental Consultant Lester from Port Coquitlam, really loves skydiving, property developers in singapore and cake decorating. that included taking a trip to Ancient City of Ping Yao.
Feel free to visit my blog; thewrightview.com
- ↑ 1.0 1.1 1.2 M. Davis and T. Januskiewicz, 1991.
- ↑ V. Buchstaber and T. Panov, 2002.
- ↑ V. Buchstaber and N. Ray, 2008.
- ↑ V. Buchstaber and N. Ray, 2001.
- ↑ V. Buchstaber, T. Panov and N. Ray, 2007.
- ↑ M. Davis and T. Januskiewicz, 1991, Proposition 1.8.
- ↑ V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.11.
- ↑ V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.12.
- ↑ Y. Civan and N. Ray, 2005.
- ↑ M. Masuda and D. Y. Suh 2007.