Wall group
An Abelian group associated with a ring with an involution which is an anti-isomorphism. In particular, it is defined for any group ring , where
is the fundamental group of a space. If
is a Poincaré complex, then for a bordism class
in
there is an obstruction in this group to the existence of a simple homotopy equivalence in
. This obstruction is called the Wall invariant, cf. [1].
Let be a ring with an involution
which is an anti-isomorphism, i.e.
. If
is a left
-module, then
is a left
-module relative to the action
,
,
,
. This module is denoted by
. For a finitely-generated projective
-module
there is an isomorphism
:
, and one may identify
and
using this isomorphism.
A quadratic -form over a ring
with an involution is a pair
, where
is a finitely-generated projective
-module and
is a homomorphism such that
. A morphism
of forms is a homomorphism
such that
. If
is an isomorphism, then the form
is said to be non-degenerate. A Lagrange plane of a non-degenerate form is a direct summand
for which
. If
is a direct summand such that
, then
is called a subLagrange plane. Two Lagrange planes
of a form
are called complementary if
and
.
Let be a projective
-module. The non-degenerate
-form
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is called Hamiltonian, and are called its complementary Lagrange planes. If
is a Lagrange plane of the form
, then the form is isomorphic to the Hamiltonian form
. The choice of a Lagrange plane complementary to
is equivalent to the choice of an isomorphism
, and this complementary plane can be identified with
.
Let be the Abelian group generated by the equivalence classes (under isomorphism) of non-degenerate quadratic
-forms
with the relations: 1)
; and 2)
if
has a Lagrange plane. A triple
consisting of a non-degenerate
-form
and a pair of Lagrange planes is called a
-formation. A formation is said to be trivial if
and
are complementary, and elementary if there exists a Lagrange plane of
which is complementary to both
and
. The trivial formation
is called Hamiltonian. By an isomorphism of formations,
, one understands an isomorphism
of forms for which
,
. Every trivial formation is isomorphic to the Hamiltonian one.
Let be the Abelian group generated by the equivalence classes (under isomorphism) of
-formations with the following relations: a)
; b)
if the formation is elementary or trivial.
The groups are called the Wall groups of the ring
.
References
[1] | C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970) |
[2] | A.A. Ranicki, "The algebraic theory of surgery I" Proc. London Math. Soc. , 40 : 1 (1980) pp. 87–192 |
Comments
In the case of and the Wall surgery obstruction invariant, the involution on
is given by
,
, where the group homomorphism
is given by the first Stiefel–Whitney class of the bundle
in the bordism class
.
The Wall groups are more often called
-groups and denoted by
; their theory is referred to as
-theory, which is much related to
-theory. (Indeed, some authors speak of the
-theory of forms, [a2].) The
-groups are four-periodic, i.e.
.
-groups can be defined in more general situations and there are a number of somewhat different varieties of
-groups, cf. e.g. [a1], [a2].
References
[a1] | A. Ranicki, "Lower ![]() ![]() |
[a2] | A. Bak, "![]() |
Wall group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wall_group&oldid=24138