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In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister (Template:Harvtxt) for 3-manifolds and generalized to higher dimensions by Template:Harvtxt and Template:Harvtxt. Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Template:Harvs as an analytic analogue of Reidemeister torsion. Template:Harvs and Template:Harvtxt proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.

Reidemeister torsion was the first invariant in algebraic topology that could distinguish between spaces which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces.

Reidemeister torsion is closely related to Whitehead torsion; see Template:Harv. For later work on torsion see the books Template:Harv, Template:Harvs. And it had given one of important motivation to arithmetic topology. Template:Harv

Definition of analytic torsion

If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the i-forms with values in E. If the eigenvalues on i-forms are λ_j then the zeta function ζ_i is defined to be

\zeta _{i}(s)=\sum _{\lambda _{j}>0}\lambda _{j}^{-s}

for s large, and this is extended to all complex s by analytic continuation. The zeta regularized determinant of the Laplacian acting on i-forms is

\Delta _{i}=\exp(-\zeta _{i}^{\prime }(0))

which is formally the product of the positive eigenvalues of the laplacian acting on i-forms. The analytic torsion T(M,E) is defined to be

T(M,E)=\exp \left(\sum _{i}(-1)^{i}i\zeta _{i}^{\prime }(0)/2\right)=\prod _{i}\Delta _{i}^{-(-1)^{i}i/2}.

Definition of Reidemeister torsion

Let $X$ be a finite connected CW-complex with fundamental group $\pi :=\pi _{1}(X)$ and $U$ an orthogonal finite-dimensional $\pi$ -representation. Suppose that

H_{n}^{\pi }(X;U):=H_{n}(U\otimes _{\mathbf {Z} [\pi ]}C_{*}({\tilde {X}}))=0

for all n. If we fix a cellular basis for $C_{*}({\tilde {X}})$ and an orthogonal $\mathbf {R}$ -basis for $U$ , then $D_{*}:=U\otimes _{\mathbf {Z} [\pi ]}C_{*}({\tilde {X}})$ is a contractible finite based free $\mathbf {R}$ -chain complex. Let $\gamma _{*}:D_{*}\to D_{*+1}$ be any chain contraction of D_*, i.e. $d_{n+1}\circ \gamma _{n}+\gamma _{n-1}\circ d_{n}=id_{D_{n}}$ for all n. We obtain an isomorphism $(d_{*}+\gamma _{*})_{odd}:D_{odd}\to D_{even}$ with $D_{odd}:=\oplus _{n\,odd}\,D_{n}$ , $D_{even}:=\oplus _{n\,even}\,D_{n}$ . We define the Reidemeister torsion

\rho (X;U):=|\mathop {det} (A)|^{-1}\in \mathbf {R} ^{>0}

where A is the matrix of $(d_{*}+\gamma _{*})_{odd}$ with respect to the given bases. The Reidemeister torsion $\rho (X;U)$ is independent of the choice of the cellular basis for $C_{*}({\tilde {X}})$ , the orthogonal basis for $U$ and the chain contraction $\gamma _{*}$ .

Let $M$ be a compact smooth manifold, and let $\rho :\pi (M)\rightarrow GL(E)$ be a unimodular representation. $M$ has a smooth triangulation. For any choice of a volume $\mu \in \mathop {det} H_{*}(M)$ , we get an invariant $\tau _{M}(\rho :\mu )\in \mathbf {R} ^{+}$ . Then we call the positive real number $\tau _{M}(\rho :\mu )$ the Reidemiester torsion of the manifold $M$ respect to $\rho$ and $\mu$ .

A short History of Reidemeister Torsion

Reidemeister torsion was first used to combinatorially classify 3-dimensional lens spaces in Template:Harv by Reidemeister, and in higher dimensional by Franz. The classification includes examples of homotopy equivalent 3-dimensional manifolds which are not homeomorphic – at the time (1935) the classification was only up to PL homeomorphism, but later Template:Harv showed that this was in fact a classification up to homeomorphism.

J. H. C. Whitehead defined the "torsion" of a homotopy equivalence between finite complexes. This is a direct generalization of the Reidemeister, Franz, and de Rham concept; but is a more delicate invariant. Whitehead torsion provides a key tool for the study of combinatorial or differentiable manifolds with nontrivial fundamental group and is closely related to the concept of "simple homotopy type." see Template:Harv

In 1960 Milnor discovered the duality relation of torsion invariants of manifolds and show that the (twisted) Alexander polynomial of knots is the Reidemister torsion of its knot complement in S³. Template:Harv For each q the Poincaré duality $P_{o}$ induces

P_{o}:\operatorname {det} (H_{q}(M)){\stackrel {\sim }{\longrightarrow }}(\operatorname {det} (H_{n-q}(M)))^{-1}

and then we obtain

\Delta (t)=\pm t^{n}\Delta (1/t).

The representation of the fundamental group of knot complement plays a central role in them. It gives the relation between knot theory and torsion invariants.

Cheeger-Müller Theorem

Let $(M,g)$ be an orientable compact Riemann manifold of dimension n and $\rho :\pi (M)\rightarrow \mathop {GL} (E)$ a representation of the fundamental group of $M$ on a real vector space of dimension N. Then we can define the De Rham complex

\Lambda ^{0}{\stackrel {d_{0}}{\longrightarrow }}\Lambda ^{1}{\stackrel {d_{1}}{\longrightarrow }}\cdots {\stackrel {d_{n-1}}{\longrightarrow }}\Lambda ^{n}

and the formal adjoint $d_{p}$ and $\delta _{p}$ due to the flatness of $E_{q}$ . And we also obtain the Laplacian on p-form as usual

\Delta _{p}=\delta _{p}d_{p}+d_{p-1}\delta _{p-1}.

We assume $\partial M=0$ , then the Laplacian is a symmetric positive simipositive elliptic operator with pure point spectrum

0\leq \lambda _{0}\leq \lambda _{1}\leq \cdots \rightarrow \infty .

As same as the above definition we can define the zeta function associated with the Laplacian $\Delta _{q}$ on $\Lambda ^{q}(E)$ by

\zeta _{q}(s;\rho )=\sum _{\lambda _{j}>0}\lambda _{j}^{-s}={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }t^{s-1}\mathop {Tr} (e^{-t\Delta _{q}}-P_{q})dt,\ \ \ \mathop {Re} (s)>{\frac {n}{2}}

where $P$ is the projection of $L^{2}\Lambda (E)$ onto the kernel space ${\mathcal {H}}^{q}(E)$ of the Laplacian $\Delta _{q}$ .

In 1967 Seeley proved that $\zeta _{q}(s;\rho )$ extends to a meromorphic function of $s\in \mathbf {C}$ which is holomorphic at $s=0$ . Template:Harv

As in the case of an orthogonal representation, we define the analytic torsion $T_{M}(\rho ;E)$ by

T_{M}(\rho ;E)=\exp {\biggl (}{\frac {1}{2}}\sum _{q=0}^{n}(-l)^{q}q{\frac {d}{ds}}\zeta _{q}(s;\rho ){\biggl |}_{s=0}{\biggr )}.

In 1971 D.B. Ray and I.M. Singer conjectured that $T_{M}(\rho ;E)=\tau _{M}(\rho ;\mu )$ for any unitary representation $\rho$ .Template:Harvs Independently, J. Cheeger Template:Harvs and W. Muller Template:Harvtxt proved the Ray-Singer conjecture. Their idea is considering the logarithm of torsions and their traces. Firstly for odd dimensional manifolds they had proved the equality of two torsions and then for even dimensional, which have some technical difficulties.

In later years, along with Atiyah-Patodi-Singer thorem, the Cheeger-Muller theorem, i.e. the equivalence of two torsions, forms the basis of Chern-Simons perturbation theory.

References

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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
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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 Online book
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
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.

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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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+In mathematics, '''Reidemeister torsion''' (or '''R-torsion''', or '''Reidemeister–Franz torsion''') is a [[topological invariant]] of [[manifold]]s introduced by [[Kurt Reidemeister]] ({{Harvtxt|Reidemeister|1935}}) for [[3-manifold]]s and generalized to higher [[dimension]]s by {{harvtxt|Franz|1935}} and {{harvtxt|de Rham|1936}}.
+'''Analytic torsion''' (or '''Ray–Singer torsion''') is an invariant of [[Riemannian manifold]]s defined by {{harvs|txt=yes|last1=Ray|last2=Singer|year1=1971|year2=1973a|year3=1973b}} as an analytic analogue of Reidemeister torsion. {{harvs|txt=yes|last=Cheeger|year1=1977|year2=1979}} and {{harvtxt|Müller|1978}} proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.
+Reidemeister torsion was the first invariant in [[algebraic topology]] that could distinguish between spaces which are [[homotopy equivalent]] but not [[homeomorphic]], and can thus be seen as the birth of [[geometric topology]] as a distinct field. It can be used to classify [[lens space]]s.
+Reidemeister torsion is closely related to [[Whitehead torsion]]; see {{harv|Milnor|1966}}. For later work on torsion see the books {{harv|Turaev|2002}}, {{harvs|last=Nicolaescu|year1=2002|year2=2003}}. And it had given one of important motivation to [[arithmetic topology]]. {{harv|Mazur|}}
+==Definition of analytic torsion==
+If ''M'' is a Riemannian manifold and ''E'' a vector bundle over ''M'', then there is a [[Laplacian operator]] acting on the ''i''-forms with values in ''E''. If the [[eigenvalue]]s on ''i''-forms are λ<sub>''j''</sub> then the zeta function ζ<sub>''i''</sub> is defined to be
+:<math>\zeta_i(s) = \sum_{\lambda_j>0}\lambda_j^{-s}</math>
+for ''s'' large, and this is extended to all complex ''s'' by [[analytic continuation]].
+The zeta regularized determinant of the Laplacian acting on ''i''-forms is
+:<math>\Delta_i=\exp(-\zeta^\prime_i(0))</math>
+which is formally the product of the positive eigenvalues of the laplacian acting on ''i''-forms.
+The '''analytic torsion''' ''T''(''M'',''E'') is defined to be
+:<math>T(M,E) = \exp\left(\sum_i (-1)^ii \zeta^\prime_i(0)/2\right) = \prod_i\Delta_i^{-(-1)^ii/2}.</math>
+==Definition of Reidemeister torsion==
+Let <math>X</math> be a finite connected [[CW complex|CW-complex]] with [[fundamental group]] <math>\pi := \pi_1(X)</math> and <math>U</math> an orthogonal finite-dimensional <math>\pi</math>-representation. Suppose that
+:<math>H^\pi_n(X;U) := H_n(U \otimes_{\mathbf{Z}[\pi]} C_*({\tilde X})) = 0</math>
+for all n. If we fix a cellular basis for <math>C_*({\tilde X})</math> and an orthogonal <math>\mathbf{R}</math>-basis for <math>U</math>, then <math>D_* := U \otimes_{\mathbf{Z}[\pi]} C_*({\tilde X})</math> is a contractible finite based free <math>\mathbf{R}</math>-chain complex. Let <math>\gamma_*: D_* \to D_{*+1}</math> be any chain contraction of D<sub>*</sub>, i.e. <math>d_{n+1} \circ \gamma_n + \gamma_{n-1} \circ d_n = id_{D_n}</math> for all n. We obtain an isomorphism <math>(d_* + \gamma_*)_{odd}: D_{odd} \to D_{even}</math> with <math>D_{odd} := \oplus_{n \, odd} \, D_n</math>, <math>D_{even} := \oplus_{n \, even} \, D_n</math>. We define the '''Reidemeister torsion'''
+:<math>\rho(X;U) := |\mathop{det}(A)|^{-1} \in \mathbf{R}^{>0}</math>
+where A is the matrix of <math>(d_* + \gamma_*)_{odd}</math> with respect to the given bases. The Reidemeister torsion <math>\rho(X;U)</math> is independent of the choice of the cellular basis for <math>C_*({\tilde X})</math>, the orthogonal basis for <math>U</math> and the chain contraction <math>\gamma_*</math>.
+Let <math>M</math> be a compact smooth manifold, and let <math>\rho:\pi(M)\rightarrow GL(E)</math> be a unimodular representation. <math>M</math> has a smooth triangulation. For any choice of a volume <math>\mu\in\mathop{det}H_*(M)</math>, we get an invariant <math>\tau_M(\rho:\mu)\in\mathbf{R}^+</math>. Then we call the positive real number <math>\tau_M(\rho:\mu)</math> the Reidemiester torsion of the manifold <math>M</math> respect to <math>\rho</math> and <math>\mu</math>.
+==A short History of Reidemeister Torsion==
+Reidemeister torsion was first used to combinatorially classify 3-dimensional [[lens space]]s in {{harv|Reidemeister|1935}} by Reidemeister, and in higher dimensional by Franz. The classification includes examples of [[homotopy equivalent]] 3-dimensional manifolds which are not [[homeomorphic]] – at the time (1935) the classification was only up to [[PL homeomorphism]], but later {{harv|Brody|1960}} showed that this was in fact a classification up to [[homeomorphism]].
+J. H. C. Whitehead defined the "torsion" of a homotopy equivalence between finite complexes. This is a direct generalization of the Reidemeister, Franz, and de Rham concept; but is a more delicate invariant. [[Whitehead torsion]] provides a key tool for the study of combinatorial or differentiable manifolds with nontrivial fundamental group and is closely related to the concept of "simple homotopy type." see {{harv|Milnor|1966}}
+In 1960 Milnor discovered the duality relation of torsion invariants of manifolds and show that the (twisted) Alexander polynomial of knots is the Reidemister torsion of its knot complement in ''S''<sup>3</sup>. {{harv|Milnor|1962}} For each ''q'' the Poincaré duality <math>P_o</math> induces
+:<math>P_o:\operatorname{det}(H_q(M))\stackrel{\sim}{\longrightarrow}(\operatorname{det}(H_{n-q}(M)))^{-1}</math>
+and then we obtain
+:<math>\Delta(t)=\pm t^n\Delta(1/t).</math>
+The representation of the fundamental group of knot complement plays a central role in them. It gives the relation between knot theory and torsion invariants.
+==Cheeger-Müller Theorem==
+Let <math>(M,g)</math> be an orientable compact Riemann manifold of dimension n and <math>\rho:\pi(M)\rightarrow\mathop{GL}(E)</math> a representation of the fundamental group of <math>M</math> on a real vector space of dimension N. Then we can define the De Rham complex
+:<math>\Lambda^0\stackrel{d_0}{\longrightarrow}\Lambda^1\stackrel{d_1}{\longrightarrow}\cdots\stackrel{d_{n-1}}{\longrightarrow}\Lambda^n</math>
+and the formal adjoint <math>d_p</math> and <math>\delta_p</math> due to the flatness of <math>E_q</math>. And we also obtain the Laplacian on p-form as usual
+:<math>\Delta_p=\delta_p d_p+d_{p-1}\delta_{p-1}.</math>
+We assume <math>\partial M=0</math>, then the Laplacian is a symmetric positive simipositive elliptic operator with pure point spectrum
+:<math>0\le\lambda_0\le\lambda_1\le\cdots\rightarrow\infty.</math>
+As same as the above definition we can define the zeta function associated with the Laplacian <math>\Delta_q</math> on <math>\Lambda^q(E)</math> by
+:<math>\zeta_q(s;\rho)=\sum_{\lambda_j >0}\lambda_j^{-s}=\frac{1}{\Gamma(s)}\int^\infty_0 t^{s-1}\mathop{Tr}(e^{-t\Delta_q} - P_q)dt,\ \ \ \mathop{Re}(s)>\frac{n}{2}</math>
+where <math>P</math> is the projection of <math>L^2 \Lambda(E)</math> onto the kernel space <math>\mathcal{H}^q(E)</math> of the Laplacian <math>\Delta_q</math>.
+In 1967 Seeley proved that <math>\zeta_q(s;\rho)</math> extends to a meromorphic function of <math>s\in\mathbf{C}</math> which is holomorphic at <math>s=0</math>. {{harv|Seeley|1967}}
+As in the case of an orthogonal representation, we define the analytic torsion <math>T_M(\rho;E)</math> by
+:<math>T_M(\rho;E) = \exp\biggl(\frac{1}{2}\sum^n_{q=0}(-l)^qq\frac{d}{ds}\zeta_q(s;\rho)\biggl|_{s=0}\biggr).</math>
+In 1971 D.B. Ray and I.M. Singer conjectured that <math>T_M(\rho;E)=\tau_M(\rho;\mu)</math> for any unitary representation <math>\rho</math>.{{harvs|txt=yes|last1=Ray|last2=Singer|year1=1971}} Independently, J. Cheeger {{harvs|txt=yes|last=Cheeger|year1=1977|year2=1979}} and W. Muller {{harvtxt|Müller|1978}} proved the Ray-Singer conjecture. Their idea is considering the logarithm of torsions and their traces. Firstly for odd dimensional manifolds they had proved the equality of two torsions and then for even dimensional, which have some technical difficulties.
+In later years, along with [[Atiyah–Singer index theorem|Atiyah-Patodi-Singer thorem]], the Cheeger-Muller theorem, i.e. the equivalence of two torsions, forms the basis of [[Chern-Simons theory|Chern-Simons perturbation theory]].
+==References==
+* {{citation| doi=10.2307/1969884| first=E. J. |last=Brody |title=The topological classification of the lens spaces| jstor=1969884|journal=[[Annals of Mathematics]] | series=2 | volume=71| issue=1 | year=1960 | pages=163–184}}
+*{{citation
+|journal=PNAS | year= 1977 | issue= 7 | pages=2651–2654
+|title=Analytic Torsion and Reidemeister Torsion
+|first=Jeff |last=Cheeger |authorlink=Jeff Cheeger
+|url=http://www.pnas.org/cgi/content/abstract/74/7/2651|doi=10.1073/pnas.74.7.2651|volume=74|pmid=16592411|pmc=431228
+|mr=0451312}}
+*{{citation
+|last=Cheeger|first= Jeff
+|title=Analytic torsion and the heat equation
+|journal=Ann. Of Math. (2) |volume=109 |year=1979|issue= 2|pages= 259–322|doi=10.2307/1971113|jstor=1971113|publisher=Annals of Mathematics
+|mr=0528965
+}}
+*{{citation|first=W.|last= Franz|title=Ueber die Torsion einer Ueberdeckung|journal=  J. Reine Angew. Math. |volume= 173  |year=1935|pages= 245–254}}
+*{{citation
+|last=Milnor|first= J.
+|title=A Duality Thorem for Reidemeister Torsion.
+|journal=Ann. of Math. |volume=76
+|issue=1 |year=1962 |pages=137–138}}
+*{{citation
+|last=Milnor|first= J.
+|title=Whitehead torsion.
+|journal=Bull. Amer. Math. Soc. |volume=72
+|issue=3 |year=1966 |pages=358–426
+|url=http://www.ams.org/bull/1966-72-03/S0002-9904-1966-11484-2/home.html|doi=10.1090/S0002-9904-1966-11484-2
+|mr=0196736}}
+*{{springer|id=R/r080960|title=Reidemeister torsion|first=A.S. |last=Mishchenko}}
+*{{citation
+|last=Müller|first= Werner
+|title=Analytic torsion and R-torsion of Riemannian manifolds.
+|journal=Adv. In Math.|volume= 28 |year=1978|issue= 3|pages=233–305
+|doi=10.1016/0001-8708(78)90116-0
+|mr=0498252 }}
+*{{citation|first=Liviu I.|last= Nicolaescu|year=2002|url=http://www.nd.edu/~lnicolae/Torsion.pdf|title=Notes on the Reidemeister torsion}} Online book
+*{{citation|last=Nicolaescu|first= Liviu I.|title= The Reidemeister torsion of 3-manifolds|series= de Gruyter Studies in Mathematics|volume= 30|publisher=Walter de Gruyter & Co.|publication-place= Berlin|year= 2003|pages= xiv+249 | isbn=3-11-017383-2|mr=1968575}}
+*{{citation
+|last= Ray|first= D. B.|last2= Singer|first2= I. M.
+|title=Analytic torsion for complex manifolds.
+|journal=Ann. Of Math. (2) |volume= 98  |year=1973a|pages= 154–177|doi=10.2307/1970909|jstor=1970909|issue=1|publisher=Annals of Mathematics
+|mr= 0383463
+}}
+*{{citation
+|last= Ray|first= D. B.|last2= Singer|first2= I. M.
+|chapter=Analytic torsion.|title= Partial differential equations |series=Proc. Sympos. Pure Math.|volume= XXIII |pages= 167–181|publisher= Amer. Math. Soc.|publication-place= Providence, R.I.|year=1973b
+|mr= 0339293}}
+*{{citation
+|last= Ray|first= D. B.|last2= Singer|first2= I. M.
+|title=''R''-torsion and the Laplacian on Riemannian manifolds.
+|journal=Advances in Math.
+|volume=7
+|issue= 2|pages=145–210|year=1971|doi=10.1016/0001-8708(71)90045-4
+|mr= 0295381 }}
+*{{citation|first=Kurt |last=Reidemeister|authorlink=Kurt Reidemeister|title=Homotopieringe und Linsenräume|journal=  Abh. Math. Sem. Univ. Hamburg |volume= 11  |year=1935|pages= 102–109|doi=10.1007/BF02940717}}
+*{{citation|first=G. |last=de Rham|title=Sur les nouveaux invariants de M. Reidemeister|journal=  Mat. Sb. ,|volume=1 |issue= 5  |year=1936|pages= 737–743}}
+*{{citation
+|last= Turaev|first= Vladimir|title= Torsions of 3-dimensional manifolds|series=Progress in Mathematics|volume= 208 |publisher=Birkhäuser Verlag|publication-place= Basel|year= 2002|pages= x+196 |ISBN= 3-7643-6911-6
+|mr= 1958479}}
+*{{citation
+| year= |title=REMARKS ON THE ALEXANDER POLYNOMIAL
+|first=Barry |last=Mazur |url=http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf }}
+*{{Citation | last1=Seeley | first1=R. T. | editor1-last=Calderón | editor1-first=Alberto P. | title=Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Proceedings of Symposia in Pure Mathematics | isbn=978-0-8218-1410-9 | mr=0237943 | year=1967 | volume=10 | chapter=Complex powers of an elliptic operator | pages=288–307}}
+[[Category:Differential geometry]]
+[[Category:3-manifolds]]
+[[Category:Surgery theory]]

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Contents

Definition of analytic torsion

Definition of Reidemeister torsion

A short History of Reidemeister Torsion

Cheeger-Müller Theorem

References

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Definition of analytic torsion

Definition of Reidemeister torsion

A short History of Reidemeister Torsion

Cheeger-Müller Theorem

References

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