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One-surgery on a three-manifold as framed knot surgery

Example

Assume ${A C}_{ω}$ , as in the surgery definition. For a framed knot in a closed oriented $3$ -manifold, $1$ -surgery replaces $S^{1} \times int D^{2}$ by $D^{2} \times S^{1}$ , with the boundary identification fixed by the framing.

For the unknot use $S^{3} = \partial (D_{x}^{2} \times D_{y}^{2})$ with compatible corner rounding, decomposed into $U = S_{x}^{1} \times D_{y}^{2}$ and $V = D_{x}^{2} \times S_{y}^{1}$ . The core is $S_{x}^{1} \times {0} \subset U$ . Its framing with integer twist $k$ is the actual product embedding $φ_{k} (x, z) = (x, x^{k} z)$ into $U$ . Zero twist produces $S^{2} \times S^{1}$ ; one twist produces $S^{3}$ . Their fundamental groups are $Z$ and $0$ , so these are different results for the same underlying knot. A bare swap of the two boundary circles is not a framing change: it exchanges the meridian with a longitude and does not extend over the removed solid torus.

Verification

Given: the unknot core in $U$ , and the framed embeddings $φ_{0}$ and $φ_{1}$ ; circle coordinates are complex numbers of modulus one.

[F1] p-surgery on a smooth m-manifold specifies the gluing by the framed product embedding on the boundary torus.

[F2] The outgoing boundary of a handle attachment trades the disk factors gives $\partial (D^{2} \times D^{2}) = (S^{1} \times D^{2}) \cup (D^{2} \times S^{1})$ , with handle parameters $k = 2$ , $n = 4$ .

[F3] $π_{1} (X \times Y, (x_{0}, y_{0})) ≅ π_{1} (X, x_{0}) \times π_{1} (Y, y_{0})$ , $Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism, and $S^{n}$ is simply connected for every $n \geq 2$ give $π_{1} (S^{2} \times S^{1}) ≅ Z$ and $π_{1} (S^{3}) = 0$ .

[F4] A diffeomorphism and its inverse give inverse induced maps on loop classes by composition (The homomorphism on fundamental groups induced by a pointed continuous map), so distinct fundamental groups rule out diffeomorphism.

1.1F1givenconstruct

Each $φ_{k}$ is a smooth embedding with inverse $(x, y) \mapsto (x, x^{- k} y)$ on $U$ , and all have core $S^{1} \times {0}$ . Writing the replacement torus as $T = D_{a}^{2} \times S_{b}^{1}$ , its boundary gluing to $V$ is $g_{k} (a, b) = (a, a^{k} b)$ . This follows directly from [F1], using $a$ as the attaching-sphere coordinate and $b$ as the normal-circle coordinate.

2.1F1step 1.1algebra

For $k = 0$ , $g_{0}$ is the product identification. Thus $V \cup_{g_{0}} T = (D^{2} \cup_{S^{1}} D^{2}) \times S^{1} ≅ S^{2} \times S^{1}$ . To check smoothness of the disk double identification, map polar disk coordinates $(r, u)$ in the two copies to $(\sin (π r / 2) u, \pm \cos (π r / 2))$ on $S^{2}$ ; it is smooth and invertible at the centres and in the signed collar coordinate at the seam.

2.2F1F2step 1.1constructalgebra

For $k = 1$ , change coordinates by diffeomorphisms of the solid tori themselves: $L : V \to V$ , $L (x, y) = (x y^{- 1}, y)$ , and $R : T \to T$ , $R (a, b) = (a b^{- 1}, b)$ . The transformed boundary gluing is $L \circ g_{1} \circ R (a, b) = (b^{- 1}, a)$ , as direct multiplication shows. This exchanges the two boundary circle factors with one reversal. In the boundary decomposition of [F2], identify the first solid torus $S^{1} \times D^{2}$ with $T = D^{2} \times S^{1}$ by $(a, b) \mapsto (b^{- 1}, a)$ ; it is a diffeomorphism, so the transformed gluing produces the boundary of $D^{2} \times D^{2}$ . A convex corner rounding is radially transverse to all rays from the origin; write its boundary as $ρ (u) u$ for smooth positive $ρ$ on $S^{3}$ . The radial map and its inverse $z \mapsto z / ∣ z ∣$ prove that boundary is diffeomorphic to $S^{3}$ . Hence one-twist surgery gives $S^{3}$ .

3.1F3F4step 2.1step 2.2∎

By [F3] and [F4], the manifolds computed in steps 2.1 and 2.2 cannot be diffeomorphic. These two valid framings of the same core knot therefore give different diffeomorphism types.

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Sources

Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002; electronic copy) (standard reference, not scraped)
C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156, Cambridge University Press 2016) (standard reference, not scraped)
Wolfgang Lück, A Basic Introduction to Surgery Theory (lecture notes, Münster, 27 October 2004) (standard reference, not scraped)