Free cash flow

In mathematics algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory', is important in surgery theory.

Definition

One can define L-groups for any ring with involution R: the quadratic L-groups $L_{*}(R)$ (Wall) and the symmetric L-groups $L^{*}(R)$ (Mishchenko, Ranicki).

Even dimension

The even dimensional L-groups $L_{2k}(R)$ are defined as the Witt groups of ε-quadratic forms over the ring R with $\epsilon =(-1)^{k}$. More precisely,

$L_{2k}(R)$

is the abelian group of equivalence classes $[\psi ]$ of non-degenerate ε-quadratic forms $\psi \in Q_{\epsilon }(F)$ over R, where the underlying R-modules F are finitely generated free. The equivalence relation is given by stabilization with respect to hyperbolic ε-quadratic forms:

$[\psi ]=[\psi ']\Longleftrightarrow n,n'\in {\mathbb {N} }_{0}:\psi \oplus H_{(-1)^{k}}(R)^{n}\cong \psi '\oplus H_{(-1)^{k}}(R)^{n'}$.

The addition in $L_{2k}(R)$ is defined by

$[\psi _{1}]+[\psi _{2}]:=[\psi _{1}\oplus \psi _{2}].$

The zero element is represented by $H_{(-1)^{k}}(R)^{n}$ for any $n\in {\mathbb {N} }_{0}$. The inverse of $[\psi ]$ is $[-\psi ]$.

Odd dimension

Defining odd dimensional L-groups is more complicated; further details and the definition of the odd dimensional L-groups can be found in the references mentioned below.

Examples and applications

The L-groups of a group $\pi$ are the L-groups $L_{*}(\mathbf {Z} [\pi ])$ of the group ring $\mathbf {Z} [\pi ]$. In the applications to topology $\pi$ is the fundamental group $\pi _{1}(X)$ of a space $X$. The quadratic L-groups $L_{*}(\mathbf {Z} [\pi ])$ play a central role in the surgery classification of the homotopy types of $n$-dimensional manifolds of dimension $n>4$, and in the formulation of the Novikov conjecture.

The distinction between symmetric L-groups and quadratic L-groups, indicated by upper and lower indices, reflects the usage in group homology and cohomology. The group cohomology $H^{*}$ of the cyclic group $\mathbf {Z} _{2}$ deals with the fixed points of a $\mathbf {Z} _{2}$-action, while the group homology $H_{*}$ deals with the orbits of a $\mathbf {Z} _{2}$-action; compare $X^{G}$ (fixed points) and $X_{G}=X/G$ (orbits, quotient) for upper/lower index notation.

The quadratic L-groups: $L_{n}(R)$ and the symmetric L-groups: $L^{n}(R)$ are related by a symmetrization map $L_{n}(R)\to L^{n}(R)$ which is an isomorphism modulo 2-torsion, and which corresponds to the polarization identities.

The quadratic L-groups are 4-fold periodic. Symmetric L-groups are not 4-periodic in general (see Ranicki, page 12), though they are for the integers.

In view of the applications to the classification of manifolds there are extensive calculations of the quadratic $L$-groups $L_{*}(\mathbf {Z} [\pi ])$. For finite $\pi$ algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite $\pi$.

More generally, one can define L-groups for any additive category with a chain duality, as in Ranicki (section 1).

Integers

The simply connected L-groups are also the L-groups of the integers, as $L(e):=L(\mathbf {Z} [e])=L(\mathbf {Z} )$ for both $L$ = $L^{*}$ or $L_{*}.$ For quadratic L-groups, these are the surgery obstructions to simply connected surgery.

The quadratic L-groups of the integers are:

${\begin{aligned}L_{4k}(\mathbf {Z} )&=\mathbf {Z} &&{\text{signature}}/8\\L_{4k+1}(\mathbf {Z} )&=0\\L_{4k+2}(\mathbf {Z} )&=\mathbf {Z} /2&&{\text{Arf invariant}}\\L_{4k+3}(\mathbf {Z} )&=0.\end{aligned}}$

In doubly even dimension (4k), the quadratic L-groups detect the signature; in singly even dimension (4k+2), the L-groups detect the Arf invariant (topologically the Kervaire invariant).

The symmetric L-groups of the integers are:

${\begin{aligned}L^{4k}(\mathbf {Z} )&=\mathbf {Z} &&{\text{signature}}\\L^{4k+1}(\mathbf {Z} )&=\mathbf {Z} /2&&{\text{de Rham invariant}}\\L^{4k+2}(\mathbf {Z} )&=0\\L^{4k+3}(\mathbf {Z} )&=0.\end{aligned}}$

In doubly even dimension (4k), the symmetric L-groups, as with the quadratic L-groups, detect the signature; in dimension (4k+1), the L-groups detect the de Rham invariant.

References

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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