World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0000696568
Reproduction Date:

Title: L-theory  
Author: World Heritage Encyclopedia
Language: English
Subject: Surgery theory, Polarization identity, L-group, C. T. C. Wall, Novikov conjecture
Collection: Algebraic Topology, Geometric Topology, Quadratic Forms, Surgery Theory
Publisher: World Heritage Encyclopedia


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 1
    • Even dimension 1.1
    • Odd dimension 1.2
  • Examples and applications 2
    • Integers 2.1
  • References 3


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


is the abelian group of equivalence classes [\psi] 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 {\mathbb N}_0: \psi \oplus H_{(-1)^k}(R)^n \cong \psi' \oplus H_{(-1)^k}(R)^{n'}.

The addition in L_{2k}(R) is defined by

[\psi_1] + [\psi_2] := [\psi_1 \oplus \psi_2].

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

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{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 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{Z}_2 deals with the fixed points of a \mathbf{Z}_2-action, while the group homology H_* deals with the orbits of a \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) \to 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. Symmetric L-groups are not 4-periodic in general (see Ranicki, page 12), though they are for the integers.

In view of the applications to the classification of manifolds there are extensive calculations of the quadratic L-groups L_*(\mathbf{Z}[\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).


The simply connected L-groups are also the L-groups of the integers, as L(e) := L(\mathbf{Z}[e]) = L(\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:

\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:

\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.


  • Lück, Wolfgang (2002), "A basic introduction to surgery theory", Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001) (PDF), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–224,  
  • Ranicki, A. A. (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics 102,  
  • Wall, C. T. C. (1999) [1970], Ranicki, Andrew, ed., Surgery on compact manifolds (PDF), Mathematical Surveys and Monographs 69 (2nd ed.), Providence, R.I.:  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.