Dehn surgery (Ex)

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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
\displaystyle \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 0-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
\displaystyle \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.


[edit] References

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