Quadratic forms for surgery

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1 Introduction

Let $(f, b) \colon M \to X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(f, b) \colon M \to X$ be a degree one normal map from a manifold of dimension $2q$$2q$. Then the surgery kernel of $(f, b)$$(f, b)$, $K_q(M)$$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

2 Topology

2.1 The 4k+2 dimensional case

Let

$\displaystyle (f,b): M \to X$ $$\displaystyle (f,b): M \to X$$

be a $q$$q$-connected normal map, $m =2q$$m =2q$, $q$$q$ odd and $q \ge 3$$q \ge 3$. Assume also $X$$X$ is $1$$1$-connected. Let

$\displaystyle I_q(M)$ $$\displaystyle I_q(M)$$

be the set of regular homotopy classes of immersions $S^q \to M$$S^q \to M$ which represent elements of the surgery kernel $K_q(M)$$K_q(M)$ (with respect to the homomorphism $I_q(M) \to K_q(M)$$I_q(M) \to K_q(M)$.

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

Then we have three invariants:

• $\mu =$$\mu =$ double point obstruction $I_q(M) \to \Bbb Z_2$$I_q(M) \to \Bbb Z_2$,

• ${\mathcal O} =$${\mathcal O} =$ Browder's framing obstruction $I_q(M) \to \Bbb Z_2$$I_q(M) \to \Bbb Z_2$, and

• $\mu' = \mu + {\mathcal O}$$\mu' = \mu + {\mathcal O}$.

Let us briefly recall how

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${\cal O}$ is defined. Each element of $x\in I_q(M)$$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$$\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$$\phi$. The homotopy class of the latter diagram defines an element of $\pi_q(\text{O}/\text{O}_q) \in \Bbb Z_2$$\pi_q(\text{O}/\text{O}_q) \in \Bbb Z_2$. This element

defines

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${\cal O}(x)$.

Theorem 2.1. The function

$\displaystyle \mu': I_q(M) \to \Bbb Z_2$ $$\displaystyle \mu': I_q(M) \to \Bbb Z_2$$

is homotopy invariant. That is, if $a,b: S^q \to M$$a,b: S^q \to M$ are immersions representing the same element $x \in K_q(M)$$x \in K_q(M)$, then $\mu'(a) = \mu'(b)$$\mu'(a) = \mu'(b)$).

Proof: The homomorphism $I_q(M) \to K_q(M)$$I_q(M) \to K_q(M)$ is onto and two-to-one. The distinct elements over a given $x \in K_q(M)$$x \in K_q(M)$ are detected by Browder's framing obstruction

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$$\displaystyle {\cal O} \in \pi_{q}(\text{O}/\text{O}_q) = \Bbb Z_2$$

(this uses Smale-Hirsch theory).

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

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${\cal O}(a) = 0$ (so

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${\cal O}(b) =1$). Then $a$$a$ is a framed immersion.

• Case 1: $\mu(a) = 0$$\mu(a) = 0$.

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

• Case 2: $\mu(a) = 1$$\mu(a) = 1$.

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

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

Corollary 2.2. The function

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${\cal O}: E_q(M) \to \Bbb Z_2$ factors through $K_q(M)$$K_q(M)$.