Kirby [Kirb 1] has developed a 'calculus' to manipulate handle decompositions of 4-manifolds which can be used to prove that two such manifolds are isomorphic. (See also [FR] and [Sa].) n = 0.

One idea is that the Kirby theorem follows from your favourite finite presentation of the mapping class group (say one of Wajnryb's), and that you can translate there and back between surgery presentations and Heegaard splittings of $3$-manifolds.

David Gay and Robion Kirby, \Trisecting 4-manifolds", Geom. Topol. 20 (2016), 3097-3132 Peter Lambert-Cole, \Bridge trisections in CP2 and the Thom Conjecture", arXiv:1807.10131 Some highlights from the course are the following three results: 1.There exist smooth homeomorphic 4-manifolds which are not di eomorphic.

Aug 8, 2018 · The Kirby calculus for such objects is well understood and documented. Each space has a handle decomposition with exactly two 2-handles. A Kirby diagram for $\mathbb{C}P^1\times\mathbb{C}P^1\cong S^2\times S^2$ consists of Hopf link of two $0$-framed unknots.

In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves.

4-manifolds as it is currently understood. It is presented from a topologist’s viewpoint, often from the perspective of handlebody theory (Kirby calculus), for which an elementary and comprehensive exposition is given.

(c) Draw a Kirby diagram of the complex projective plane CP2. Exercise 3. (a) Let F be a closed oriented surface with a handle decomposition with exactly one 0-handle and exactly one 2-handle. (c) Describe the 4-torus T4 = S1 S1 S1 S1 as a Kirby diagram. Exercise 4.

CP2 is the complex projective plane and —CP2 is CP2 with the opposite orientation ; they are described by a 0-handle, a 2-handle attached to the unknot with framing ± 1, and a 4-handle; the 0 and 2-handles give the Hopf or antiHopf disk bundle with boundary S3.

Content: The goal of Kirby calculus, named after the American mathematician Robion Kirby, is to study smooth compact 4-manifolds by splitting them into simple pieces. This simple pieces, so-called handles, are (after smoothing the corners) all diffeomorphic to 4-balls.

Kirby Diagrams and Framings Exercises Exercise #1: Let X be a simply-connected 4-manifold with nonempty connected boundary. Show that X is determined by X 2 and the number of 3-handles attached. Figure 1: Knot in S1 S2 Exercise #2: (Exercise 4.4.4) For the knot K in S1 S2 shown above, why algebraically does