This page has not been refereed. The information given here might be incomplete or provisional.
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1 Introduction
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This is a work in progress! Initial blurb.
Let be an oriented cover of a connected manifold and let , be -trivial immersions of manifolds in . The equivariant intersection number counts with elements of the number of intersection points that the two immersions have. The intersection number is an obstruction to perturbing the immersions into being disjoint, which when zero can often be achieved using the Whitney trick.
The intersection number is also used in defining the intersection form of a -dimensional manifold and in turn its signature - a very important invariant used in the classification of manifolds and the primary surgery obstruction.
2 Definition
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The
homology intersection pairing of
with respect to an oriented cover
is the sesquilinear pairing defined by
with
the Poincaré dual of
with respect to Universal Poincaré duality, such that
The
algebraic intersection number of
-trivial maps
,
with prescribed lifts
,
is the homology intersection of the homology classes
,
:
3 Alternative Description: Lifts
As in the non-equivariant case the equivariant intersection form has a geometric interpretation. Let be an oriented cover of a connected manifold with -twisted fundamental class corresponding to the lift of the basepoint . Let , be transverse -trivial immersions of oriented manifolds with prescribed lifts , .
At a double point
there is a unique covering translation
such that
The lifted immersions
,
have a transverse double point
and there is defined an isomorphism of oriented
-dimensional vector spaces
where
,
and
all inherit orientations from the given orientations of
,
and
.
The
equivariant index of a transverse double point
is defined to be
with
The
geometric equivariant intersection number of
and
is then defined to be
The effect on the equivariant index of a change of order in the double point is given by
with
the
-twisted involution
as
and the orientation of
disagrees withe the orientation of
if and only if
and
are both odd. Consequently
Observe that
agrees with the non-equivariant index
of the transverse double point
from which it follows that
4 Alternative Description: Paths
Let
,
and
such that
be basepoints. Then there is a bijective correspondence
as explained in the page on
-trivial maps. Thus there are two equivalent conventions we can use for the data of a
-trivial map: either a choice of lift or a choice of path from
to
modulo
. Both conventions have equivalent definitions for the equivariant intersection number of transversely intersecting
-trivial immersions.
Let
,
be
-trivial immersions with prescribed equivalence classes of paths
such that
and
for
. At a transverse double point
define
as the loop
where
is any path from
to
. This loop is well-defined since a different choice of representative of
or a different path
results in a loop that differs from the other by an element of
or
which is trivial in
.
Definition of :
Equivalence:
Let be a basepoint of , a basepoint of for and let be some choice of lift. For a transverse double point , an isotopy class of paths from to corresponds to a lift as follows.
The geometric intersection number of transverse immersions is
5 Equivalence of definitions
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The algebraic and geometric intersection numbers agree. See REFERENCE
6 Examples
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...
7 References