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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){displaystyle L_{*}(R)} (Wall) and the symmetric L-groups L∗(R){displaystyle L^{*}(R)} (Mishchenko, Ranicki).

Even dimension

The even-dimensional L-groups L2k(R){displaystyle L_{2k}(R)} are defined as the Witt groups of ε-quadratic forms over the ring R with ϵ=(−1)k{displaystyle epsilon =(-1)^{k}}. More precisely,

L2k(R){displaystyle L_{2k}(R)}

is the abelian group of equivalence classes [ψ]{displaystyle [psi ]} of non-degenerate ε-quadratic forms ψ∈Qϵ(F){displaystyle 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:

[ψ]=[ψ′]⟺n,n′∈N0:ψ⊕H(−1)k(R)n≅ψ′⊕H(−1)k(R)n′{displaystyle [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 L2k(R){displaystyle L_{2k}(R)} is defined by

[ψ1]+[ψ2]:=[ψ1⊕ψ2].{displaystyle [psi _{1}]+[psi _{2}]:=[psi _{1}oplus psi _{2}].}

The zero element is represented by H(−1)k(R)n{displaystyle H_{(-1)^{k}}(R)^{n}} for any n∈N0{displaystyle nin {mathbb {N} }_{0}}. The inverse of [ψ]{displaystyle [psi ]} is [−ψ]{displaystyle [-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 π{displaystyle pi } are the L-groups L∗(Z[π]){displaystyle L_{*}(mathbf {Z} [pi ])} of the group ring Z[π]{displaystyle mathbf {Z} [pi ]}. In the applications to topology π{displaystyle pi } is the fundamental group
π1(X){displaystyle pi _{1}(X)} of a space X{displaystyle X}. The quadratic L-groups L∗(Z[π]){displaystyle L_{*}(mathbf {Z} [pi ])}
play a central role in the surgery classification of the homotopy types of n{displaystyle n}-dimensional manifolds of dimension n>4{displaystyle 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∗{displaystyle H^{*}} of the cyclic group Z2{displaystyle mathbf {Z} _{2}} deals with the fixed points of a Z2{displaystyle mathbf {Z} _{2}}-action, while the group homology H∗{displaystyle H_{*}} deals with the orbits of a Z2{displaystyle mathbf {Z} _{2}}-action; compare XG{displaystyle X^{G}} (fixed points) and XG=X/G{displaystyle X_{G}=X/G} (orbits, quotient) for upper/lower index notation.

The quadratic L-groups: Ln(R){displaystyle L_{n}(R)} and the symmetric L-groups: Ln(R){displaystyle L^{n}(R)} are related by
a symmetrization map Ln(R)→Ln(R){displaystyle 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{displaystyle L}-groups L∗(Z[π]){displaystyle L_{*}(mathbf {Z} [pi ])}. For finite π{displaystyle pi }
algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite π{displaystyle 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(Z[e])=L(Z){displaystyle L(e):=L(mathbf {Z} [e])=L(mathbf {Z} )} for both L{displaystyle L} = L∗{displaystyle L^{*}} or L∗.{displaystyle L_{*}.} For quadratic L-groups, these are the surgery obstructions to simply connected surgery.

The quadratic L-groups of the integers are:

L4k(Z)=Zsignature/8L4k+1(Z)=0L4k+2(Z)=Z/2Arf invariantL4k+3(Z)=0.{displaystyle {begin{aligned}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{aligned}}}

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:

L4k(Z)=ZsignatureL4k+1(Z)=Z/2de Rham invariantL4k+2(Z)=0L4k+3(Z)=0.{displaystyle {begin{aligned}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{aligned}}}

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.

References

• 
  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, MR 1937016.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Ranicki, Andrew A. (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics, 102, Cambridge University Press, ISBN 978-0-521-42024-2, MR 1211640
  


• 
  Wall, C. T. C. (1999) [1970], Ranicki, Andrew, ed., Surgery on compact manifolds (PDF), Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388