# 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 $\mathbf Z [ \pi _ {1} ( X)]$, 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

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].