L-theory

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 even-dimensional L-groups $L\_\{2k\}(R)$ are defined as the Witt groups of ε-quadratic forms over the ring R with $\backslash epsilon\; =\; (-1)^k$. More precisely,

::$L\_\{2k\}(R)$

is the abelian group of equivalence classes $[\backslash psi]$ of non-degenerate ε-quadratic forms $\backslash psi\; \backslash in\; Q\_\backslash epsilon(F)$ 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:

:$[\backslash psi]\; =\; [\backslash psi\text{'}]\; \backslash Longleftrightarrow\; n,\; n\text{'}\; \backslash in\; \{\backslash mathbb\; N\}\_0:\; \backslash psi\; \backslash oplus\; H\_\{(-1)^k\}(R)^n\; \backslash cong\; \backslash psi\text{'}\; \backslash oplus\; H\_\{(-1)^k\}(R)^\{n\text{'}\}$.

The addition in $L\_\{2k\}(R)$ is defined by

:$[\backslash psi\_1]\; +\; [\backslash psi\_2]\; :=\; [\backslash psi\_1\; \backslash oplus\; \backslash psi\_2].$

The zero element is represented by $H\_\{(-1)^k\}(R)^n$ for any $n\; \backslash in\; \{\backslash mathbb\; N\}\_0$. The inverse of $[\backslash psi]$ is $[-\backslash psi]$.

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.

The L-groups of a group $\backslash pi$ are the L-groups $L\_*(\backslash mathbf\{Z\}[\backslash pi])$ of the group ring $\backslash mathbf\{Z\}[\backslash pi]$. In the applications to topology $\backslash pi$ is the fundamental group

$\backslash pi\_1\; (X)$ of a space $X$. The quadratic L-groups $L\_*(\backslash mathbf\{Z\}[\backslash pi])$

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 $\backslash mathbf\{Z\}\_2$ deals with the fixed points of a $\backslash mathbf\{Z\}\_2$-action, while the group homology $H\_*$ deals with the orbits of a $\backslash mathbf\{Z\}\_2$-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 $L\_n(R)\; \backslash to\; L^n(R)$ 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 $L\_*(\backslash mathbf\{Z\}[\backslash pi])$. For finite $\backslash pi$

algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite $\backslash pi$.

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

The simply connected L-groups are also the L-groups of the integers, as

$L(e)\; :=\; L(\backslash mathbf\{Z\}[e])\; =\; L(\backslash mathbf\{Z\})$ 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:

:$\backslash begin\{align\}$

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}

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:

:$\backslash begin\{align\}$

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}

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.

{{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