Quadratic forms for surgery

From Manifold Atlas
Jump to: navigation, search

This page has not been refereed. The information given here might be incomplete or provisional.


[edit] 1 Introduction

Let (f, b) \colon M \to X be a degree one normal map from a manifold of dimension 2q. Then the surgery kernel of (f, b), K_q(M), 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

\displaystyle (f,b): M \to X
be a q-connected normal map, m =2q, q odd and q \ge 3. Assume also X is 1-connected. Let
\displaystyle I_q(M)
be the set of regular homotopy classes of immersions S^q \to M which represent elements of the surgery kernel K_q(M) (with respect to the homomorphism I_q(M) \to K_q(M).

It is an abelian group using the connected summ operation (this uses the condition q \ge 3).

Then we have three invariants:

  • \mu = double point obstruction I_q(M) \to  \Bbb Z_2,
  • {\mathcal O} = Browder's framing obstruction I_q(M) \to  \Bbb Z_2, and
  • \mu' = \mu + {\mathcal O}.

[edit] 1 Definition of the framing obstruction

Each element of x\in I_q(M) is represented by a commutative square

\displaystyle  \SelectTips{cm}{} \xymatrix{ S^q \ar[r]^\phi \ar[d] & M \ar[d]^{f}\\ D^{q+1} \ar[r] & X }

with \phi an immersion, and a diagram of normal bundle data

\displaystyle  \SelectTips{cm}{} \xymatrix{ S^q \ar[r]^{\nu_\phi} \ar[d] & B\text{O}_q \ar[d]^{f}\\ D^{q+1} \ar[r] & B\text{O} }

the latter defining a stable trivialization of the normal bundle of \phi. The homotopy class of the latter diagram defines an element of \pi_q(\text{O}/\text{O}_q) \in \Bbb Z_2. This element defines {\mathcal O}(x).

[edit] 2 Definition of the self-intersection obstruction

A generic immersion \phi: S^q \to M has only double points with transverse crossings. Then \mu([\phi]) \in {\Bbb Z}_2 is defined to be the number of double points of \phi taken modulo two. This only depends on [\phi]= the regular homotopy class of \phi.

Note that \mu is a quadratic function, i.e.,

\displaystyle  \mu(x+y) = \mu(x) + \mu(y) + x\cdot y

where x\cdot y denotes the intersection pairing applied to x and y considered as elements of H_q(M;{\Bbb Z}_2).

[edit] 3 Homotopy Invariance

Theorem 2.1. The function
\displaystyle  \mu': I_q(M) \to  \Bbb Z_2
is homotopy invariant. That is, if a,b: S^q \to  M are immersions representing the same element x \in K_q(M), then \mu'(a) = \mu'(b)).

Proof: The homomorphism I_q(M) \to  K_q(M) is onto and two-to-one. The distinct elements over a given x \in K_q(M) are detected by Browder's framing obstruction
\displaystyle {\mathcal O} \in \pi_{q}(\text{O}/\text{O}_q) = \Bbb Z_2

(this uses Smale-Hirsch theory).

Let a and b be immersions representing these elements. Then a and b are not regularly homotopic. (Note: when q\ne 3,7 the normal bundles of a and b are distinct; when q=3,7 they are both trivial.) We can assume without loss in generality that {\mathcal O}(a) = 0 (so {\mathcal O}(b) =1). Then a is a framed immersion.

  • Case 1: \mu(a) = 0.

If \mu(a) = 0 then the Whitney trick shows that a is regularly homotopic to a (framed) embedding, so assume that a is a framed embedding. Whitney's method of introducing a single double point to a in a coordinate chart yields a new immersion b' such that b' has one double point and b' still represents x. Then \mu(b') = 1, so b' isn't regularly homotopic to a. It must therefore be regularly homotopic to b. Hence \mu(b) = 1. It follows that \mu'(a) = \mu'(b) in this case.

  • Case 2: \mu(a) = 1.

In this case a is regularly homotopic to an immersion with exactly one double point. By introducing another double point we get a b'' representing x such that \mu(b'') = 0. Then b'' is not regularly homotopic to a so it must be regularly homotopic to b. Consequently, \mu(b) = 0. Therefore \mu'(a) = \mu'(b) in this case.\Box

Let E_q(M) denote the isotopy classes of embeddings S^q \to  M representing elements of K_q(M). Then we have a function E_q(M) \to  I_q(M).

Corollary 2.2. The function {\mathcal O}: E_q(M)  \to  \Bbb Z_2 factors through K_q(M).

[edit] 3 Algebra

[edit] 4 References

Personal tools