Equivariant intersection number of π-trivial immersions
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[edit] 1 Introduction
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.
[edit] 2 Definition
[edit] 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 , .
[edit] 4 Alternative Description: Paths
Thus there are two equivalent conventions for the data of a -trivial map : either a choice of lift of or the homotopy class of a choice of path from to modulo . In the previous section the geometric equivariant intersection number of and was defined using lifts as data. In this section we see the equivalent approach of using paths and prove the equivalence.
Let , be -trivial immersions with prescribed equivalence classes of paths such that and for . At a transverse double point define to be the class of the loop
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
[edit] 5 Equivalence of definitions
The algebraic and geometric intersection numbers agree. See REFERENCE
[edit] 6 Examples
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