View source for Wall invariant

<!--
w0970301.png
$#A+1 = 51 n = 1
$#C+1 = 51 : ~/encyclopedia/old_files/data/W097/W.0907030 Wall invariant
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

An element of the [[Wall group]], representing the [[Obstruction|obstruction]] to the surgery of a [[bordism]] to a simple homotopy equivalence.

Let  $  X $
be a finite [[Poincaré complex|Poincaré complex]],  $  \nu $
a fibre bundle over  $  X $
and  $  x = [( M, \phi , F  )] \in \Omega ( X, \nu ) $
a bordism class, where  $  m $
is the formal dimension of  $  X $
and  $  \phi :  M \rightarrow X $
has degree 1. This mapping can always be represented by an  $  [ m/2] $-
connected mapping using a finite sequence of surgeries. Let  $  \Lambda = Z [ \pi _ {1} ( X)] $
be a group ring and let  $   \overline{ {}}\;  $
be the involution on  $  \Lambda $
given by the formula  $  \overline{ {\sum _ {g} n ( g) g }}\; = \sum w ( g) n ( g) g  ^ {-1} $, 
where  $  w:  \pi _ {1} ( X) \rightarrow \{ 1, - 1 \} $
is defined by the first [[Stiefel–Whitney class]] of  $  \nu $. 
Put

$$ 
K  ^ {*} ( M)  = \ 
 \mathop{\rm coker} ( \phi  ^ {*} :  H  ^ {*} ( X) \rightarrow H  ^ {*} ( M)),
$$

$$ 
K _ {*} ( M)  =   \mathop{\rm ker} ( \phi _ {*} :  H _ {*} ( M) \rightarrow H _ {*} ( X))
$$

(coefficients in  $  \Lambda $). 
The involution is an anti-isomorphism and there is defined the Wall group  $  U _ {n} ( \Lambda ) = L _ {n} ( \pi _ {1} ( X), w) $.

Suppose now that  $  m = 2k \geq  4 $. 
Then in the stable free  $  \Lambda $-
module  $  G = K _ {k} ( M) = \pi _ {k + 1 }  ( \phi ) $
one can choose a basis, and [[Poincaré duality|Poincaré duality]] induces a simple isomorphism  $  \lambda :  G \rightarrow G  ^ {*} = K  ^ {k} ( M) $, 
where  $  ( G, \lambda ) $
is a  $  (- 1)  ^ {k} $-
form. One therefore obtains the class  $  \Theta _ {2k} ( x) = [( G, \lambda )] \in L _ {2k} ( \pi _ {1} ( X), w) $.

Suppose next that  $  m = 2k + 1 \geq  5 $. 
One can choose generators in  $  \pi _ {k + 1 }  ( \phi ) = K _ {k} ( M;  \Lambda ) $
so that they represent the imbeddings  $  f _ {i} :  S  ^ {k} \times D ^ {k + 1 } \rightarrow M $, 
with non-intersecting images, and these images are connected by paths with a base point. Put  $  U = \cup _ {i}  \mathop{\rm Im}  f _ {i} $, 
$  M _ {0} = M \setminus   \mathop{\rm Int}  U $. 
Since  $  \phi \circ f _ {i} \sim 0 $, 
one may replace  $  \phi $
by a homotopy and suppose that  $  \phi ( u) = * $. 
Because  $  X $
is a Poincaré complex, one can replace  $  X $
by a complex with a unique  $  m $-
cell, i.e. one has a Poincaré pair  $  ( X _ {0} , S ^ {m + 1 } ) $
and  $  X = X _ {0} \cup e  ^ {m} $. 
By the choice of a suitable cellular approximation one obtains a mapping for the Poincaré triad of degree 1:  $  \phi :  ( M;  M _ {0} , U) \rightarrow ( X;  X _ {0} , e  ^ {m} ) $. 
Consequently one has the diagram of exact sequences:

$$

Moreover, one has a non-degenerate pairing  $  \lambda :  K _ {k} ( \partial  U) \times K _ {k} ( \partial  U) \rightarrow \Lambda $, 
where  $  H = ( K _ {k} ( \partial  U), \lambda ) $
is a quadratic  $  (- 1)  ^ {k} $-
form while  $  K _ {k + 1 }  ( U, \partial  U) $
and  $  K _ {k + 1 }  ( M _ {0} , \partial  U) $
define its Lagrange planes  $  L $
and  $  P $. 
Then  $  \Theta _ {2k + 1 }  ( x) = [( H;  L, P)] \in U _ {2k + 1 }  ( \Lambda ) = L _ {2k + 1 }  ( \pi _ {1} ( x), w) $.

The elements  $  \Theta _ {m} ( x) \in L _ {m} ( \pi _ {1} ( x), w) $
defined above are called the Wall invariants. An important property is the independence of  $  \Theta ( x) $
on the choices in the construction and the equivalence of the equation  $  \Theta ( x) = 0 $
to the representability of the class as a simple homotopy equivalence, cf. [[#References|[1]]].

====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><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Novikov,   "Algebraic construction and properties of Hermitian analogs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703052.png" />-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes I"  ''Math. USSR Izv.'' , '''4''' :  2  (1970)  pp. 257–292  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34''' :  2  (1970)  pp. 253–288</TD></TR></table>