p/q Dehn surgery on knots produces 3-manifolds. Explore lens spaces, S³, and Heegaard diagrams.
Dehn Surgery
Surgery: remove tubular
neighborhood, reglue solid
torus by (p,q) homeomorphism
Surgery: 1/1
Manifold: L(1,1) = S³
π₁: trivial
H₁: 0
Manifold Classification
Lens space L(1,1) ≅ S³
The 3-sphere. p/q = 1/1 surgery on the unknot gives back S³.
Dehn surgery theorem (Lickorish-Wallace, 1960-1): Every closed orientable 3-manifold is obtained by integral Dehn surgery on a link in S³.
Construction: Remove a tubular neighborhood N(K) ≅ T²×[0,1] of knot K. Reglue solid torus D²×S¹ by attaching the meridian
to the curve p[μ]+q[λ] on ∂N(K), where μ = meridian, λ = longitude.
Unknot: p/q surgery gives lens space L(p,q). In particular: 0/1 → S²×S¹; 1/0 → S³; p/1 → L(p,1).
Trefoil: +1 surgery → Poincaré homology sphere (π₁ = binary icosahedral group, H₁=0, but not S³!).