Wall invariant
An element of the Wall group, representing the obstruction to the surgery of a bordism to a simple homotopy equivalence.
Let be a finite Poincaré complex,
a fibre bundle over
and
a bordism class, where
is the formal dimension of
and
has degree 1. This mapping can always be represented by an
-connected mapping using a finite sequence of surgeries. Let
be a group ring and let
be the involution on
given by the formula
, where
is defined by the first Stiefel–Whitney class of
. Put
![]() |
![]() |
(coefficients in ). The involution is an anti-isomorphism and there is defined the Wall group
.
Suppose now that . Then in the stable free
-module
one can choose a basis, and Poincaré duality induces a simple isomorphism
, where
is a
-form. One therefore obtains the class
.
Suppose next that . One can choose generators in
so that they represent the imbeddings
, with non-intersecting images, and these images are connected by paths with a base point. Put
,
. Since
, one may replace
by a homotopy and suppose that
. Because
is a Poincaré complex, one can replace
by a complex with a unique
-cell, i.e. one has a Poincaré pair
and
. By the choice of a suitable cellular approximation one obtains a mapping for the Poincaré triad of degree 1:
. Consequently one has the diagram of exact sequences:
![]() |
Moreover, one has a non-degenerate pairing , where
is a quadratic
-form while
and
define its Lagrange planes
and
. Then
.
The elements defined above are called the Wall invariants. An important property is the independence of
on the choices in the construction and the equivalence of the equation
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 ![]() |
Wall invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wall_invariant&oldid=17462