L-theory

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 very 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).

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 manifoldsof dimension $n>\; 4$.

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

The distinction between symmetric "L"-groups and quadratic "L"-groups, indicated by upper and lower indices, reflectsthe 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 $H\_*$ dealswith the orbits of a $mathbf\{Z\}\_2$-action.

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

The quadratic "L"-groups are 4-fold periodic. The quadratic "L"-groups of the integers are::

Symmetric "L"-groups are not 4-periodic in general (p. 12),though they are for the integers.The symmetric "L"-groups of the integers are::

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

External links

* [http://www.maths.ed.ac.uk/~aar/books/scm.pdf Surgery on compact manifolds] , by C.T.C. Wall.
* [http://www.maths.ed.ac.uk/~aar/books/topman.pdf Algebraic "L"-theory and topological manifolds] , by Andrew Ranicki.