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| In [[mathematics]] algebraic '''L-theory''' is the [[K-theory]] of [[quadratic form]]s; the term was coined by [[C. T. C. Wall]],
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| with ''L'' being used as the letter after ''K''. Algebraic ''L''-theory, also known as 'hermitian ''K''-theory',
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| is important in [[surgery theory]].
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| ==Definition==
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| 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).
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| === Even dimension ===
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| 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,
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| <math>L_{2k}(R)</math>
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| 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]]:
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| :<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>.
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| The addition in <math>L_{2k}(R)</math> is defined by
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| :<math>[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2].</math>
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| 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>.
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| === Odd dimension ===
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| 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.
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| ==Examples and applications==
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| 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]]
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| <math>\pi_1 (X)</math> of a space <math>X</math>. The quadratic ''L''-groups <math>L_*(\mathbf{Z}[\pi])</math>
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| 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]].
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| 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.
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| The quadratic ''L''-groups: <math>L_n(R)</math> and the symmetric ''L''-groups: <math>L^n(R)</math> are related by
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| 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]].
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| 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.
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| In view of the applications to the [[classification of manifolds]] there are extensive calculations of
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| the quadratic <math>L</math>-groups <math>L_*(\mathbf{Z}[\pi])</math>. For finite <math>\pi</math>
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| algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite <math>\pi</math>.
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| More generally, one can define ''L''-groups for any [[additive category]] with a ''chain duality'', as in Ranicki (section 1).
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| === Integers ===
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| The '''simply connected ''L''-groups''' are also the ''L''-groups of the integers, as
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| <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.
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| The quadratic ''L''-groups of the integers are:
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| :<math>\begin{align}
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| L_{4k}(\mathbf{Z}) &= \mathbf{Z} && \text{signature}/8\\
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| L_{4k+1}(\mathbf{Z}) &= 0\\
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| L_{4k+2}(\mathbf{Z}) &= \mathbf{Z}/2 && \text{Arf invariant}\\
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| L_{4k+3}(\mathbf{Z}) &= 0.
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| \end{align}</math>
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| 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]]).
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| The symmetric ''L''-groups of the integers are:
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| :<math>\begin{align}
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| L^{4k}(\mathbf{Z}) &= \mathbf{Z} && \text{signature}\\
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| L^{4k+1}(\mathbf{Z}) &= \mathbf{Z}/2 && \text{de Rham invariant}\\
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| L^{4k+2}(\mathbf{Z}) &= 0\\
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| L^{4k+3}(\mathbf{Z}) &= 0.
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| \end{align}</math>
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| 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]].
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| ==References==
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| [[Category:Geometric topology]]
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| [[Category:Algebraic topology]]
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| [[Category:Quadratic forms]]
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| [[Category:Surgery theory]]
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I'm a 47 years old, married and work at the high school (Dramatic Literature and History).
In my free time I try to learn French. I have been there and look forward to go there sometime near future. I love to read, preferably on my beloved Kindle. I like to watch 2 Broke Girls and Doctor Who as well as docus about anything geological. I enjoy Table football.
Here is my webpage - stop coming too early