Editing L-theory

{{DISPLAYTITLE:''L''-theory}}
In [[mathematics]], algebraic '''''L''-theory''' is the [[K-theory|''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]].<ref>{{cite web|url=https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.582711 |title=L-theory, K-theory and involutions, by Levikov, Filipp, 2013, On University of Aberdeen(ISNI:0000 0004 2745 8820)}}</ref>

==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).

=== 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 and the symmetric ''L''-groups are 4-fold periodic (the comment of Ranicki, page 12, on the non-periodicity of the symmetric ''L''-groups refers to another type of ''L''-groups, defined using "short complexes").

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==
{{Reflist}}
*{{Citation | last1=Lück | first1=Wolfgang | author-link=Wolfgang Lück |title=Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001) | chapter-url=https://www.him.uni-bonn.de/lueck/data/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=Andrew A. |author-link=Andrew Ranicki| 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. |authorlink1=C. T. C. Wall| editor1-last=Ranicki | editor1-first=Andrew | editor1-link=Andrew Ranicki|title=Surgery on compact manifolds | orig-year=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]]