Dehn surgery (Ex)

Let $K$$<wikitex>; Let $K$ be an embedded $S^1\hookrightarrow S^3$ with a closed tubular neighbourhood $\nu K \cong S^1 \times D^2$. A '''Dehn surgery''' on $K$ is the process of removing int$(\nu K)$ and gluing back in a copy of $S^1\times D^2$ by any diffeomorphism $$\phi: S^1\times \partial D^2 \to \partial\nu K$$ of the boundary tori. Orienting $K$, let $\alpha$ be a right-handed meridean and $\beta\in H_1(\partial\nu K; \Z)$ a <script type="math/tex">K$ be an embedded $S^1\hookrightarrow S^3$$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$.

References

$-framed copy of $K$ pushed to the boundary of $\nu K$. A Lens space $L(p,-q)$ is defined to be the effect of Dehn surgery on the standard embedding $S^1\hookrightarrow S^3$ with $\phi$ such that $$\phi_*([\partial D^2]) = p\alpha + q\beta.$$ '''1)''' Show $L(\pm 2, 1) \simeq \R P^3$, $L(\pm 1, 1) = L(p,0) = S^3$. '''2)''' Prove the `slam dunk' - that the combined effect of the two surgeries on the Hopf link in $S^3$ with framings $m$ and $n$ on the respective components is the Lens space $L(1-mn,n)$. Hence show that any Lens space is null-cobordant. '''Hint:''' It may help to prove that $L(p,-q) = L(-p,q)$ so that we can unambiguously consider the Dehn surgery generating the space as `$p/q$-surgery' on the embedded $S^1$. == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]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$.