Quadratic forms for surgery
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Contents |
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
2 Topology
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
- .
Tex syntax erroris defined. 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
definesTex syntax error.
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(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 thatTex syntax error(so
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- Case 1: .
If then a result of Whitney (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 .
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