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An element of the [[Wall group|Wall group]], representing the [[Obstruction|obstruction]] to the surgery of a [[Bordism|bordism]] to a simple homotopy equivalence.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970301.png" /> be a finite [[Poincaré complex|Poincaré complex]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970302.png" /> a fibre bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970304.png" /> a bordism class, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970305.png" /> is the formal dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970307.png" /> has degree 1. This mapping can always be represented by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970308.png" />-connected mapping using a finite sequence of surgeries. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w0970309.png" /> be a group ring and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703010.png" /> be the involution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703011.png" /> given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703013.png" /> is defined by the first [[Stiefel–Whitney class|Stiefel–Whitney class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703014.png" />. Put
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<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/w097030/w09703015.png" /></td> </tr></table>
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An element of the [[Wall group]], representing the [[Obstruction|obstruction]] to the surgery of a [[bordism]] to a simple homotopy equivalence.
  
<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/w097030/w09703016.png" /></td> </tr></table>
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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
  
(coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703017.png" />). The involution is an anti-isomorphism and there is defined the Wall group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703018.png" />.
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$$
 +
K  ^ {*} ( M)  = \
 +
\mathop{\rm coker} ( \phi  ^ {*} : H  ^ {*} ( X) \rightarrow H  ^ {*} ( M)),
 +
$$
  
Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703019.png" />. Then in the stable free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703020.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703021.png" /> one can choose a basis, and [[Poincaré duality|Poincaré duality]] induces a simple isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703023.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703024.png" />-form. One therefore obtains the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703025.png" />.
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$$
 +
K _ {*} ( M)  =   \mathop{\rm ker} ( \phi _ {*} : H _ {*} ( M) \rightarrow H _ {*} ( X))
 +
$$
  
Suppose next that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703026.png" />. One can choose generators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703027.png" /> so that they represent the imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703028.png" />, with non-intersecting images, and these images are connected by paths with a base point. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703030.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703031.png" />, one may replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703032.png" /> by a homotopy and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703033.png" />. Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703034.png" /> is a Poincaré complex, one can replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703035.png" /> by a complex with a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703036.png" />-cell, i.e. one has a Poincaré pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703038.png" />. By the choice of a suitable cellular approximation one obtains a mapping for the Poincaré triad of degree 1: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703039.png" />. Consequently one has the diagram of exact sequences:
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(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) $.
  
<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/w097030/w09703040.png" /></td> </tr></table>
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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) $.
  
Moreover, one has a non-degenerate pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703042.png" /> is a quadratic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703043.png" />-form while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703045.png" /> define its Lagrange planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703047.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703048.png" />.
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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:
  
The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703049.png" /> defined above are called the Wall invariants. An important property is the independence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703050.png" /> on the choices in the construction and the equivalence of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097030/w09703051.png" /> to the representability of the class as a simple homotopy equivalence, cf. [[#References|[1]]].
+
$$
 +
 
 +
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====
 
====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>
 
<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>

Latest revision as of 20:21, 16 January 2024


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

Let $ X $ be a finite 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 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. [1].

References

[1] C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970)
[2] A.A. Ranicki, "The algebraic theory of surgery I" Proc. London Math. Soc. , 40 : 1 (1980) pp. 87–192
[3] S.P. Novikov, "Algebraic construction and properties of Hermitian analogs of -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
How to Cite This Entry:
Wall invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wall_invariant&oldid=17462
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article