Difference between revisions of "Wall group"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970) {{MR|0431216}} {{ZBL|0219.57024}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Ranicki, "The algebraic theory of surgery I" ''Proc. London Math. Soc.'' , '''40''' : 1 (1980) pp. 87–192 {{MR|0560997}} {{MR|0566491}} {{ZBL|0471.57010}} </TD></TR></table> |
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Ranicki, "Lower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020106.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020107.png" />-theory" , Cambridge Univ. Press (1992) {{MR|1208729}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Bak, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020108.png" />-theory of forms" , Princeton Univ. Press (1981) {{MR|0632404}} {{ZBL|0465.10013}} </TD></TR></table> |
Revision as of 17:35, 31 March 2012
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
![]() |
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) MR0431216 Zbl 0219.57024 |
[2] | A.A. Ranicki, "The algebraic theory of surgery I" Proc. London Math. Soc. , 40 : 1 (1980) pp. 87–192 MR0560997 MR0566491 Zbl 0471.57010 |
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=11262