- Source: 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
)
{\displaystyle L_{*}(R)}
(Wall) and the symmetric L-groups
L
∗
(
R
)
{\displaystyle L^{*}(R)}
(Mishchenko, Ranicki).
= Even dimension
=The even-dimensional L-groups
L
2
k
(
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,
L
2
k
(
R
)
{\displaystyle L_{2k}(R)}
is the abelian group of equivalence classes
[
ψ
]
{\displaystyle [\psi ]}
of non-degenerate ε-quadratic forms
ψ
∈
Q
ϵ
(
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
′
∈
N
0
:
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
L
2
k
(
R
)
{\displaystyle L_{2k}(R)}
is defined by
[
ψ
1
]
+
[
ψ
2
]
:=
[
ψ
1
⊕
ψ
2
]
.
{\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
∈
N
0
{\displaystyle n\in {\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
π
1
(
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
Z
2
{\displaystyle \mathbf {Z} _{2}}
deals with the fixed points of a
Z
2
{\displaystyle \mathbf {Z} _{2}}
-action, while the group homology
H
∗
{\displaystyle H_{*}}
deals with the orbits of a
Z
2
{\displaystyle \mathbf {Z} _{2}}
-action; compare
X
G
{\displaystyle X^{G}}
(fixed points) and
X
G
=
X
/
G
{\displaystyle X_{G}=X/G}
(orbits, quotient) for upper/lower index notation.
The quadratic L-groups:
L
n
(
R
)
{\displaystyle L_{n}(R)}
and the symmetric L-groups:
L
n
(
R
)
{\displaystyle L^{n}(R)}
are related by
a symmetrization map
L
n
(
R
)
→
L
n
(
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 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
{\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:
L
4
k
(
Z
)
=
Z
signature
/
8
L
4
k
+
1
(
Z
)
=
0
L
4
k
+
2
(
Z
)
=
Z
/
2
Arf invariant
L
4
k
+
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:
L
4
k
(
Z
)
=
Z
signature
L
4
k
+
1
(
Z
)
=
Z
/
2
de Rham invariant
L
4
k
+
2
(
Z
)
=
0
L
4
k
+
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" (PDF), Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes, vol. 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 1–224, MR 1937016
Ranicki, Andrew A. (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics, vol. 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, vol. 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388
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