Namespaces
Variants
Actions

Difference between revisions of "Wall group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (MR/ZBL numbers added)
Line 18: Line 18:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.T.C. Wall,   "Surgery on compact manifolds" , Acad. Press (1970)</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</TD></TR></table>
+
<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>
  
  
Line 28: Line 28:
  
 
====References====
 
====References====
<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)</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)</TD></TR></table>
+
<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 - and -theory" , Cambridge Univ. Press (1992) MR1208729
[a2] A. Bak, "-theory of forms" , Princeton Univ. Press (1981) MR0632404 Zbl 0465.10013
How to Cite This Entry:
Wall group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wall_group&oldid=24138
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article