Quadratic forms for surgery
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Introduction
Let be a degree one normal map from a manifold of dimension . Then the surgery kernel of , , comes equipped with a subtle and crucial quadratic refinement. This page describes both the algebraic and geometric aspects of such quadratic refinements
[edit] 2 Topology
[edit] 2.1 The 4k+2 dimensional case
It is an abelian group using the connected summ operation (this uses the condition ).
Then we have three invariants:
- double point obstruction ,
- Browder's framing obstruction , and
- .
[edit] 1 Definition of the framing obstruction
Each element of is represented by a commutative square
with an immersion, and a diagram of normal bundle data
the latter defining a stable trivialization of the normal bundle of . The homotopy class of the latter diagram defines an element of . This element defines .
[edit] 2 Definition of the self-intersection obstruction
A generic immersion has only double points with transverse crossings. Then is defined to be the number of double points of taken modulo two. This only depends on the regular homotopy class of .
Note that is a quadratic function, i.e.,
where denotes the intersection pairing applied to and considered as elements of .
[edit] 3 Homotopy Invariance
(this uses Smale-Hirsch theory).
Let and be immersions representing these elements. Then and are not regularly homotopic. (Note: when the normal bundles of and are distinct; when they are both trivial.) We can assume without loss in generality that (so ). Then is a framed immersion.
- Case 1: .
If then the Whitney trick shows that is regularly homotopic to a (framed) embedding, so assume that is a framed embedding. Whitney's method of introducing a single double point to in a coordinate chart yields a new immersion such that has one double point and still represents . Then , so isn't regularly homotopic to . It must therefore be regularly homotopic to . Hence . It follows that in this case.
- Case 2: .
In this case is regularly homotopic to an immersion with exactly one double point. By introducing another double point we get a representing such that . Then is not regularly homotopic to so it must be regularly homotopic to . Consequently, . Therefore in this case.
Let denote the isotopy classes of embeddings representing elements of . Then we have a function .