Dehn surgery (Ex)

Let $K$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}K$ be an embedded $S^1\hookrightarrow S^3$$S^1\hookrightarrow S^3$ with a closed tubular neighbourhood $\nu K \cong S^1 \times D^2$$\nu K \cong S^1 \times D^2$. A Dehn surgery on $K$$K$ is the process of removing int$(\nu K)$$(\nu K)$ and gluing back in a copy of $S^1\times D^2$$S^1\times D^2$ by any diffeomorphism

$\displaystyle \phi: S^1\times \partial D^2 \to \partial\nu K$ $$\displaystyle \phi: S^1\times \partial D^2 \to \partial\nu K$$

of the boundary tori. Orienting $K$$K$, let $\alpha$$\alpha$ be a right-handed meridean and $\beta\in H_1(\partial\nu K; \Z)$$\beta\in H_1(\partial\nu K; \Z)$ a $0$$0$-framed copy of $K$$K$ pushed to the boundary of $\nu K$$\nu K$. A Lens space $L(p,-q)$$L(p,-q)$ is defined to be the effect of Dehn surgery on the standard embedding $S^1\hookrightarrow S^3$$S^1\hookrightarrow S^3$ with $\phi$$\phi$ such that

$\displaystyle \phi_*([\partial D^2]) = p\alpha + q\beta.$ $$\displaystyle \phi_*([\partial D^2]) = p\alpha + q\beta.$$

1) Show $L(\pm 2, 1) \simeq \R P^3$$L(\pm 2, 1) \simeq \R P^3$, $L(\pm 1, 1) = L(p,0) = S^3$$L(\pm 1, 1) = L(p,0) = S^3$.

2) Prove the `slam dunk' - that the combined effect of the two surgeries on the Hopf link in $S^3$$S^3$ with framings $m$$m$ and $n$$n$ on the respective components is the Lens space $L(1-mn,n)$$L(1-mn,n)$. Hence show that any Lens space is null-cobordant.

Hint: It may help to prove that $L(p,-q) = L(-p,q)$$L(p,-q) = L(-p,q)$ so that we can unambiguously consider the Dehn surgery generating the space as `$p/q$$p/q$-surgery' on the embedded $S^1$$S^1$.