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An Abelian group associated with a ring with an involution which is an anti-isomorphism. In particular, it is defined for any group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970202.png" /> is the fundamental group of a space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970203.png" /> is a [[Poincaré complex|Poincaré complex]], then for a [[Bordism|bordism]] class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970204.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970205.png" /> there is an [[Obstruction|obstruction]] in this group to the existence of a simple homotopy equivalence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970206.png" />. This obstruction is called the [[Wall invariant|Wall invariant]], cf. [[#References|[1]]].
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970207.png" /> be a ring with an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970208.png" /> which is an anti-isomorphism, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w0970209.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702010.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702011.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702012.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702013.png" />-module relative to the action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702017.png" />. This module is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702018.png" />. For a finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702019.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702020.png" /> there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702021.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702022.png" />, and one may identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702024.png" /> using this isomorphism.
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A quadratic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702025.png" />-form over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702026.png" /> with an involution is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702028.png" /> is a finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702029.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702030.png" /> is a homomorphism such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702031.png" />. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702032.png" /> of forms is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702035.png" /> is an isomorphism, then the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702036.png" /> is said to be non-degenerate. A Lagrange plane of a non-degenerate form is a direct summand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702037.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702039.png" /> is a direct summand such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702041.png" /> is called a subLagrange plane. Two Lagrange planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702042.png" /> of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702043.png" /> are called complementary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702045.png" />.
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An Abelian group associated with a ring with an involution which is an anti-isomorphism. In particular, it is defined for any group ring  $  \mathbf Z [ \pi _ {1} ( X)] $,  
 +
where $  \pi _ {1} ( X) $
 +
is the fundamental group of a space. If $  X $
 +
is a [[Poincaré complex|Poincaré complex]], then for a [[Bordism|bordism]] class  $  \alpha $
 +
in  $  \Omega _ {*} ( x, \nu ) $
 +
there is an [[Obstruction|obstruction]] in this group to the existence of a simple homotopy equivalence in  $  \alpha $.  
 +
This obstruction is called the [[Wall invariant|Wall invariant]], cf. [[#References|[1]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702046.png" /> be a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702047.png" />-module. The non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702048.png" />-form
+
Let $  R $
 +
be a ring with an involution  $  R \rightarrow R $
 +
which is an anti-isomorphism, i.e. $  \overline{ {ab }}\; = \overline{ {ba }}\; $.  
 +
If  $  P $
 +
is a left  $  R $-
 +
module, then  $  \mathop{\rm Hom} _ {R} ( P, R) $
 +
is a left  $  R $-
 +
module relative to the action  $  ( af  ) ( x) = f ( x) \overline{a}\; $,
 +
$  f \in  \mathop{\rm Hom} _ {R} ( P, R) $,
 +
$  a \in R $,
 +
$  x \in P $.  
 +
This module is denoted by  $  P  ^ {*} $.  
 +
For a finitely-generated projective  $  R $-
 +
module  $  P $
 +
there is an isomorphism  $  P \rightarrow P  ^ {**} $:
 +
$  x \mapsto ( f \mapsto \overline{ {f ( x) }}\; ) $,
 +
and one may identify  $  P $
 +
and  $  P  ^ {**} $
 +
using this isomorphism.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702049.png" /></td> </tr></table>
+
A quadratic  $  (- 1)  ^ {k} $-
 +
form over a ring  $  R $
 +
with an involution is a pair  $  ( P, \phi ) $,
 +
where  $  P $
 +
is a finitely-generated projective  $  R $-
 +
module and  $  \phi : P \rightarrow P  ^ {*} $
 +
is a homomorphism such that  $  \phi = (- 1)  ^ {k} \phi  ^ {*} $.
 +
A morphism  $  f: ( P, \phi ) \rightarrow ( Q, \psi ) $
 +
of forms is a homomorphism  $  f: P \rightarrow Q $
 +
such that  $  f ^ { * } \psi f = \phi $.
 +
If  $  \phi $
 +
is an isomorphism, then the form  $  ( P, \phi ) $
 +
is said to be non-degenerate. A Lagrange plane of a non-degenerate form is a direct summand  $  L \subset  P $
 +
for which  $  L = \mathop{\rm Ann}  \phi ( L) $.  
 +
If  $  L \subset  P $
 +
is a direct summand such that  $  L \subset  \mathop{\rm Ann}  \phi ( L) $,
 +
then  $  L $
 +
is called a subLagrange plane. Two Lagrange planes  $  L, G $
 +
of a form  $  ( P, \phi ) $
 +
are called complementary if  $  L + G = P $
 +
and  $  L \cap G = \{ 0 \} $.
  
is called Hamiltonian, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702050.png" /> are called its complementary Lagrange planes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702051.png" /> is a Lagrange plane of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702052.png" />, then the form is isomorphic to the Hamiltonian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702053.png" />. The choice of a Lagrange plane complementary to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702054.png" /> is equivalent to the choice of an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702055.png" />, and this complementary plane can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702056.png" />.
+
Let  $  L $
 +
be a projective  $  R $-
 +
module. The non-degenerate  $  (- 1)  ^ {k} $-
 +
form
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702057.png" /> be the Abelian group generated by the equivalence classes (under isomorphism) of non-degenerate quadratic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702058.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702059.png" /> with the relations: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702060.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702061.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702062.png" /> has a Lagrange plane. A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702063.png" /> consisting of a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702064.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702065.png" /> and a pair of Lagrange planes is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702067.png" />-formation. A formation is said to be trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702070.png" /> are complementary, and elementary if there exists a Lagrange plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702072.png" /> which is complementary to both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702074.png" />. The trivial formation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702075.png" /> is called Hamiltonian. By an isomorphism of formations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702077.png" />, one understands an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702078.png" /> of forms for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702080.png" />. Every trivial formation is isomorphic to the Hamiltonian one.
+
$$
 +
H _ {(- 1) ^ {k}  } ( L) = \
 +
\left ( L \oplus L  ^ {*} , \left (
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702081.png" /> be the Abelian group generated by the equivalence classes (under isomorphism) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702082.png" />-formations with the following relations: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702083.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702084.png" /> if the formation is elementary or trivial.
+
is called Hamiltonian, and  $  L, L  ^ {*} \subset  L \oplus L  ^ {*} $
 +
are called its complementary Lagrange planes. If  $  L $
 +
is a Lagrange plane of the form  $  ( P, \phi ) $,
 +
then the form is isomorphic to the Hamiltonian form  $  H _ {(- 1)  ^ {k}  } ( L) $.
 +
The choice of a Lagrange plane complementary to  $  L $
 +
is equivalent to the choice of an isomorphism $  ( P, \phi ) \rightarrow H _ {(- 1) ^ {k}  } ( L) $,
 +
and this complementary plane can be identified with  $  L  ^ {*} $.
  
The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702085.png" /> are called the Wall groups of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702086.png" />.
+
Let  $  U _ {2k} ( R ) $
 +
be the Abelian group generated by the equivalence classes (under isomorphism) of non-degenerate quadratic  $  (- 1)  ^ {k} $-
 +
forms  $  ( P, \phi ) $
 +
with the relations: 1)  $  [( P, \phi )] + [( Q, \psi )] = [( P \oplus Q, \phi \oplus \psi )] $;
 +
and 2)  $  [( P, \phi )] = 0 $
 +
if  $  P $
 +
has a Lagrange plane. A triple  $  ( H;  F, L) $
 +
consisting of a non-degenerate  $  (- 1)  ^ {k} $-
 +
form  $  H $
 +
and a pair of Lagrange planes is called a  $  (- 1)  ^ {k} $-
 +
formation. A formation is said to be trivial if  $  F $
 +
and  $  L $
 +
are complementary, and elementary if there exists a Lagrange plane of  $  H $
 +
which is complementary to both  $  F $
 +
and  $  L $.
 +
The trivial formation  $  ( H _ {(- 1)  ^ {k}  } ( G);  G, G) $
 +
is called Hamiltonian. By an isomorphism of formations,  $  f:  ( H;  F, L) \rightarrow ( H _ {1} ;  F _ {1} , L _ {1} ) $,
 +
one understands an isomorphism  $  f:  H \rightarrow H _ {1} $
 +
of forms for which  $  f ( F  ) = F _ {1} $,
 +
$  f ( L) = L _ {1} $.  
 +
Every trivial formation is isomorphic to the Hamiltonian one.
 +
 
 +
Let  $  U _ {2k + 1 }  ( R ) $
 +
be the Abelian group generated by the equivalence classes (under isomorphism) of  $  (- 1)  ^ {k} $-
 +
formations with the following relations: a)  $  [( H;  F, L)] \oplus [( H _ {1} ;  F _ {1} , L _ {1} )] = [( H \oplus H _ {1} ;  F \oplus F _ {1} , L \oplus L _ {1} )] $;
 +
b)  $  [( H;  F, L)] = 0 $
 +
if the formation is elementary or trivial.
 +
 
 +
The groups  $  U _ {n} ( R) $
 +
are called the Wall groups of the ring $  R $.
  
 
====References====
 
====References====
 
<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>
 
<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>
 
 
  
 
====Comments====
 
====Comments====
In the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702087.png" /> and the Wall surgery obstruction invariant, the involution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702088.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702090.png" />, where the group homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702091.png" /> is given by the first Stiefel–Whitney class of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702092.png" /> in the bordism class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702093.png" />.
+
In the case of $  R = \mathbf Z [ \pi _ {1} ( X) ] $
 +
and the Wall surgery obstruction invariant, the involution on $  R $
 +
is given by $  g \mapsto w( g) g  ^ {-} 1 $,  
 +
$  g \in \pi _ {1} ( X) $,  
 +
where the group homomorphism $  w : \pi _ {1} ( X) \rightarrow \{ 1, - 1 \} $
 +
is given by the first Stiefel–Whitney class of the bundle $  \nu $
 +
in the bordism class $  \Omega _ {*} ( X, \nu ) $.
  
The Wall groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702094.png" /> are more often called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702096.png" />-groups and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702097.png" />; their theory is referred to as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w09702099.png" />-theory, which is much related to [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020100.png" />-theory]]. (Indeed, some authors speak of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020101.png" />-theory of forms, [[#References|[a2]]].) The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020102.png" />-groups are four-periodic, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020103.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020104.png" />-groups can be defined in more general situations and there are a number of somewhat different varieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097020/w097020105.png" />-groups, cf. e.g. [[#References|[a1]]], [[#References|[a2]]].
+
The Wall groups $  U _ {n} ( R) $
 +
are more often called $  L $-
 +
groups and denoted by $  L _ {n} ( R) $;  
 +
their theory is referred to as $  L $-
 +
theory, which is much related to [[K-theory| $  K $-
 +
theory]]. (Indeed, some authors speak of the $  K $-
 +
theory of forms, [[#References|[a2]]].) The $  L $-
 +
groups are four-periodic, i.e. $  L _ {n} ( R) \simeq L _ {n+} 4 ( R) $.  
 +
$  L $-
 +
groups can be defined in more general situations and there are a number of somewhat different varieties of $  L $-
 +
groups, cf. e.g. [[#References|[a1]]], [[#References|[a2]]].
  
 
====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) {{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>
 
<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>

Latest revision as of 08:28, 6 June 2020


An Abelian group associated with a ring with an involution which is an anti-isomorphism. In particular, it is defined for any group ring $ \mathbf Z [ \pi _ {1} ( X)] $, where $ \pi _ {1} ( X) $ is the fundamental group of a space. If $ X $ is a Poincaré complex, then for a bordism class $ \alpha $ in $ \Omega _ {*} ( x, \nu ) $ there is an obstruction in this group to the existence of a simple homotopy equivalence in $ \alpha $. This obstruction is called the Wall invariant, cf. [1].

Let $ R $ be a ring with an involution $ R \rightarrow R $ which is an anti-isomorphism, i.e. $ \overline{ {ab }}\; = \overline{ {ba }}\; $. If $ P $ is a left $ R $- module, then $ \mathop{\rm Hom} _ {R} ( P, R) $ is a left $ R $- module relative to the action $ ( af ) ( x) = f ( x) \overline{a}\; $, $ f \in \mathop{\rm Hom} _ {R} ( P, R) $, $ a \in R $, $ x \in P $. This module is denoted by $ P ^ {*} $. For a finitely-generated projective $ R $- module $ P $ there is an isomorphism $ P \rightarrow P ^ {**} $: $ x \mapsto ( f \mapsto \overline{ {f ( x) }}\; ) $, and one may identify $ P $ and $ P ^ {**} $ using this isomorphism.

A quadratic $ (- 1) ^ {k} $- form over a ring $ R $ with an involution is a pair $ ( P, \phi ) $, where $ P $ is a finitely-generated projective $ R $- module and $ \phi : P \rightarrow P ^ {*} $ is a homomorphism such that $ \phi = (- 1) ^ {k} \phi ^ {*} $. A morphism $ f: ( P, \phi ) \rightarrow ( Q, \psi ) $ of forms is a homomorphism $ f: P \rightarrow Q $ such that $ f ^ { * } \psi f = \phi $. If $ \phi $ is an isomorphism, then the form $ ( P, \phi ) $ is said to be non-degenerate. A Lagrange plane of a non-degenerate form is a direct summand $ L \subset P $ for which $ L = \mathop{\rm Ann} \phi ( L) $. If $ L \subset P $ is a direct summand such that $ L \subset \mathop{\rm Ann} \phi ( L) $, then $ L $ is called a subLagrange plane. Two Lagrange planes $ L, G $ of a form $ ( P, \phi ) $ are called complementary if $ L + G = P $ and $ L \cap G = \{ 0 \} $.

Let $ L $ be a projective $ R $- module. The non-degenerate $ (- 1) ^ {k} $- form

$$ H _ {(- 1) ^ {k} } ( L) = \ \left ( L \oplus L ^ {*} , \left (

is called Hamiltonian, and $ L, L ^ {*} \subset L \oplus L ^ {*} $ are called its complementary Lagrange planes. If $ L $ is a Lagrange plane of the form $ ( P, \phi ) $, then the form is isomorphic to the Hamiltonian form $ H _ {(- 1) ^ {k} } ( L) $. The choice of a Lagrange plane complementary to $ L $ is equivalent to the choice of an isomorphism $ ( P, \phi ) \rightarrow H _ {(- 1) ^ {k} } ( L) $, and this complementary plane can be identified with $ L ^ {*} $.

Let $ U _ {2k} ( R ) $ be the Abelian group generated by the equivalence classes (under isomorphism) of non-degenerate quadratic $ (- 1) ^ {k} $- forms $ ( P, \phi ) $ with the relations: 1) $ [( P, \phi )] + [( Q, \psi )] = [( P \oplus Q, \phi \oplus \psi )] $; and 2) $ [( P, \phi )] = 0 $ if $ P $ has a Lagrange plane. A triple $ ( H; F, L) $ consisting of a non-degenerate $ (- 1) ^ {k} $- form $ H $ and a pair of Lagrange planes is called a $ (- 1) ^ {k} $- formation. A formation is said to be trivial if $ F $ and $ L $ are complementary, and elementary if there exists a Lagrange plane of $ H $ which is complementary to both $ F $ and $ L $. The trivial formation $ ( H _ {(- 1) ^ {k} } ( G); G, G) $ is called Hamiltonian. By an isomorphism of formations, $ f: ( H; F, L) \rightarrow ( H _ {1} ; F _ {1} , L _ {1} ) $, one understands an isomorphism $ f: H \rightarrow H _ {1} $ of forms for which $ f ( F ) = F _ {1} $, $ f ( L) = L _ {1} $. Every trivial formation is isomorphic to the Hamiltonian one.

Let $ U _ {2k + 1 } ( R ) $ be the Abelian group generated by the equivalence classes (under isomorphism) of $ (- 1) ^ {k} $- formations with the following relations: a) $ [( H; F, L)] \oplus [( H _ {1} ; F _ {1} , L _ {1} )] = [( H \oplus H _ {1} ; F \oplus F _ {1} , L \oplus L _ {1} )] $; b) $ [( H; F, L)] = 0 $ if the formation is elementary or trivial.

The groups $ U _ {n} ( R) $ are called the Wall groups of the ring $ R $.

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 $ R = \mathbf Z [ \pi _ {1} ( X) ] $ and the Wall surgery obstruction invariant, the involution on $ R $ is given by $ g \mapsto w( g) g ^ {-} 1 $, $ g \in \pi _ {1} ( X) $, where the group homomorphism $ w : \pi _ {1} ( X) \rightarrow \{ 1, - 1 \} $ is given by the first Stiefel–Whitney class of the bundle $ \nu $ in the bordism class $ \Omega _ {*} ( X, \nu ) $.

The Wall groups $ U _ {n} ( R) $ are more often called $ L $- groups and denoted by $ L _ {n} ( R) $; their theory is referred to as $ L $- theory, which is much related to $ K $- theory. (Indeed, some authors speak of the $ K $- theory of forms, [a2].) The $ L $- groups are four-periodic, i.e. $ L _ {n} ( R) \simeq L _ {n+} 4 ( R) $. $ L $- groups can be defined in more general situations and there are a number of somewhat different varieties of $ L $- 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=49167
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article