L-theory

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

::$L_(R)$

is the abelian group of equivalence classes $$psi/math> 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 _0: \psi \oplus H_(R)^n \cong \psi' \oplus H_(R)^.

The addition in $L_(R)$ is defined by

: psi_1+ psi_2:= psi_1 \oplus \psi_2

The zero element is represented by $H_(R)^n$ for any $n \in _0$. The inverse of $$psi/math> is \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 $\pi$ are the ''L''-groups L_*(\mathbf pi of the group ring \mathbf pi/math>. In the applications to topology $\pi$ is the fundamental group
$\pi_1 (X)$ of a space $X$. The quadratic ''L''-groups L_*(\mathbf 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 $\mathbf_2$ deals with the fixed points of a $\mathbf_2$-action, while the group homology $H_*$ deals with the orbits of a $\mathbf_2$-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 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 $L$-groups L_*(\mathbf 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 = L(\mathbf) 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:
:$\begin L_(\mathbf) &= \mathbf && \text/8\\ L_(\mathbf) &= 0\\ L_(\mathbf) &= \mathbf/2 && \text\\ L_(\mathbf) &= 0. \end$
In doubly even dimension (4''k''), the quadratic ''L''-groups detect the 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:
:$\begin L^(\mathbf) &= \mathbf && \text\\ L^(\mathbf) &= \mathbf/2 && \text\\ L^(\mathbf) &= 0\\ L^(\mathbf) &= 0. \end$
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

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*{{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

Geometric topology
Algebraic topology
Quadratic forms
Surgery theory