Free cash flow: Difference between revisions

Revision as of 12:02, 10 January 2014

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

$L_{2 k} (R)$

is the abelian group of equivalence classes $[ψ]$ of non-degenerate ε-quadratic forms $ψ \in Q_{ϵ} (F)$ 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:

[ψ] = [ψ^{'}] ⟺ n, n^{'} \in ℕ_{0} : ψ \oplus H_{(- 1)^{k}} (R)^{n} ≅ ψ^{'} \oplus H_{(- 1)^{k}} (R)^{n^{'}}

.

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

[ψ_{1}] + [ψ_{2}] : = [ψ_{1} \oplus ψ_{2}] .

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

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 $π$ are the L-groups $L_{*} (Z [π])$ of the group ring $Z [π]$ . In the applications to topology $π$ is the fundamental group $π_{1} (X)$ of a space $X$ . The quadratic L-groups $L_{*} (Z [π])$ 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 $Z_{2}$ deals with the fixed points of a $Z_{2}$ -action, while the group homology $H_{*}$ deals with the orbits of a $Z_{2}$ -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 $L_{*} (Z [π])$ . For finite $π$ algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite $π$ .

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 $L (e) : = L (Z [e]) = L (Z)$ 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_{4 k} (Z) & = Z & signature / 8 \\ L_{4 k + 1} (Z) & = 0 \\ L_{4 k + 2} (Z) & = Z / 2 & Arf invariant \\ L_{4 k + 3} (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^{4 k} (Z) & = Z & signature \\ L^{4 k + 1} (Z) & = Z / 2 & de Rham invariant \\ L^{4 k + 2} (Z) & = 0 \\ L^{4 k + 3} (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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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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@@ Line 1: / Line 1: @@
+In [[mathematics]] algebraic '''L-theory''' is the [[K-theory]] of [[quadratic form]]s; 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 <math>L_*(R)</math> (Wall) and the symmetric ''L''-groups <math>L^*(R)</math> (Mishchenko, Ranicki).
-Internet marketing is the best Korean blogger in his methods of website linking niche. Implementing methods of website linking successful SEO Techniques require's an extensive knowledge of logic and a very good football goal. There are two ways in which SEO doesn't exist.   [http://www.insiderpages.com/b/15260235383/orlando-seo-gaba-marketing-inc-orlando seo orlando] So it's like Google passes the authority of the link, which increases conversion rates. If your company's online image has been unfairly affected by a Rip Off Report, please contact and we can get your site ranking and avoid getting your site in droves.<br><br>Also visit my blog post ... [http://www.insiderpages.com/b/15260235383/orlando-seo-gaba-marketing-inc-orlando seo orlando]
+=== Even dimension ===
+The even dimensional ''L''-groups <math>L_{2k}(R)</math> are defined as the [[Witt group]]s of [[ε-quadratic forms]] over the ring ''R'' with <math>\epsilon = (-1)^k</math>. More precisely,
+<math>L_{2k}(R)</math>
+is the abelian group of equivalence classes <math>[\psi]</math> of non-degenerate ε-quadratic forms <math>\psi \in Q_\epsilon(F)</math> 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]]:
+:<math>[\psi] = [\psi'] \Longleftrightarrow n, n' \in {\mathbb N}_0: \psi \oplus H_{(-1)^k}(R)^n \cong \psi' \oplus H_{(-1)^k}(R)^{n'}</math>.
+The addition in <math>L_{2k}(R)</math> is defined by
+:<math>[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2].</math>
+The zero element is represented by <math>H_{(-1)^k}(R)^n</math> for any <math>n \in {\mathbb N}_0</math>. The inverse of <math>[\psi]</math> is <math>[-\psi]</math>.
+=== 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 <math>\pi</math> are the ''L''-groups <math>L_*(\mathbf{Z}[\pi])</math> of the [[group ring]] <math>\mathbf{Z}[\pi]</math>. In the applications to topology  <math>\pi</math> is the [[fundamental group]]
+<math>\pi_1 (X)</math> of a space <math>X</math>. The quadratic ''L''-groups <math>L_*(\mathbf{Z}[\pi])</math>
+play a central role in the surgery classification of the homotopy types of <math>n</math>-dimensional [[manifolds]] of dimension <math>n > 4</math>, 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]] <math>H^*</math> of the cyclic group <math>\mathbf{Z}_2</math> deals with the fixed points of a <math>\mathbf{Z}_2</math>-action, while the [[group homology]] <math>H_*</math> deals with the orbits of a <math>\mathbf{Z}_2</math>-action; compare <math>X^G</math> (fixed points) and <math>X_G = X/G</math> (orbits, quotient) for upper/lower index notation.
+The quadratic ''L''-groups: <math>L_n(R)</math> and the symmetric ''L''-groups: <math>L^n(R)</math> are related by
+a symmetrization map <math>L_n(R) \to L^n(R)</math> 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 <math>L</math>-groups <math>L_*(\mathbf{Z}[\pi])</math>. For finite <math>\pi</math>
+algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite <math>\pi</math>.
+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
+<math>L(e) := L(\mathbf{Z}[e]) = L(\mathbf{Z})</math> for both <math>L</math> = <math>L^*</math> or <math>L_*.</math> For quadratic ''L''-groups, these are the surgery obstructions to [[simply connected]] surgery.
+The quadratic ''L''-groups of the integers are:
+:<math>\begin{align}
+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{align}</math>
+In [[doubly even]] dimension (4''k''), the quadratic ''L''-groups detect the [[signature (topology)|signature]]; in [[singly even]] dimension (4''k''+2), the ''L''-groups detect the [[Arf invariant]] (topologically the [[Kervaire invariant]]).
+The symmetric ''L''-groups of the integers are:
+:<math>\begin{align}
+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{align}</math>
+In doubly even dimension (4''k''), the symmetric ''L''-groups, as with the quadratic ''L''-groups, detect the signature; in dimension (4''k''+1), the ''L''-groups detect the [[de Rham invariant]].
+==References==
+*{{Citation | last1=Lück | first1=Wolfgang | title=Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001) | url=http://www.math.uni-muenster.de/u/lueck/publ/lueck/ictp.pdf | publisher=Abdus Salam Int. Cent. Theoret. Phys., Trieste | series=ICTP Lect. Notes | mr=1937016 | year=2002 | volume=9 | chapter=A basic introduction to surgery theory | pages=1–224}}
+*{{Citation | last1=Ranicki | first1=A. A. | title=Algebraic L-theory and topological manifolds | url=http://www.maths.ed.ac.uk/~aar/books/topman.pdf | publisher=[[Cambridge University Press]] | series=Cambridge Tracts in Mathematics | isbn=978-0-521-42024-2 | mr=1211640 | year=1992 | volume=102}}
+*{{Citation | last1=Wall | first1=C. T. C. | editor1-last=Ranicki | editor1-first=Andrew | title=Surgery on compact manifolds | origyear=1970 | url=http://www.maths.ed.ac.uk/~aar/books/scm.pdf | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=Mathematical Surveys and Monographs | isbn=978-0-8218-0942-6 | mr=1687388 | year=1999 | volume=69}}
+[[Category:Geometric topology]]
+[[Category:Algebraic topology]]
+[[Category:Quadratic forms]]
+[[Category:Surgery theory]]

Free cash flow: Difference between revisions

Revision as of 12:02, 10 January 2014

Contents

Definition

Even dimension

Odd dimension

Examples and applications

Integers

References

Navigation menu

Free cash flow: Difference between revisions

Revision as of 12:02, 10 January 2014

Definition

Even dimension

Odd dimension

Examples and applications

Integers

References

Navigation menu

Search