Kirby diagrams (stub)

Abstract. A brief note about Kirby diagrams, which provide a helpful tool in the representation of 3 and 4-dimensional manifolds

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This is a note to myself. Kirby diagrams are cool. (This note is also a stub.)

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A Kirby diagram depicts

• A single 0-handle $D^4$ with boundary $S^3$; up to a compactifying point this is the background $\mathbb{R}^3$
• Attaching 1-handles $D^1 \times D^3$ with attaching disks $S^0 \times D^3$, which is the same as removing (the cancelling) 2-handle: thus we depict is as an $S^1 \times D^2$ in the boundary (with the bulk of the handle being cut out from the bulk of the manifold)
• Attaching 2-handles is shown by attaching $S^1 \times D^2$ (as $S^1$ with a number indicated the twists). We can always homotope the attachement to be in the “visible part” of the diagram, i.e., not the boundaries resulting from cutting in the bulk.

(Re last claim: this works because of 1-homotopy groups at plays—the same doesn’t hold on dimension down, where after attaching a 1-handle to a 0-handle you get a torus boundary; which has 1-homotopy $Z_2 \times Z_2$.)

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For manifold with boundary, 0-, 1-, 2-handles may be enough (e.g. for Mazur manifold). For closed manifold, we certainly need more handles. It turns out, that as long as the boundary is of the right type (see Scorpan’s book), attachment of 3- and 4- handles is unique to close up the manifold. This is why we don’t depict them.

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• Boxes with number $k$ means “twist these wires 360 degree $k$ times.