Skip to main content

# Questions tagged [kirby-diagram]

The tag has no usage guidance.

24 questions
Filter by
Sorted by
Tagged with
3 votes
0 answers
98 views

### Calculating homology of cobordism of 3-manifolds from Kirby diagram

I've been reading Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci and Stipsicz, and have gotten stuck on the following exercise on p. 44. Below $Y_1, Y_2$ are closed 3-manifolds, and $Q$ ...
• 3,405
2 votes
1 answer
43 views

### A 4-dimensional 3-handle attachment to $S^2 \times D^2$ to get $D^4$.

Given $Y= S^1\times D^2$ and attaching sphere of 3-handle as $S= S^2 \times \{pt\}$, then how to visualise that after attaching 3-handle we get $D^4$ as the resultant manifold? This is part of the ...
3 votes
1 answer
56 views

### Showing a 4-manifold is contractible

In Gompf-Stipsicz book, we're presented with the Akbulut cork, and a brief explanation of why it is contractible (see below). Would someone be able to explain what homotopy he is referring too? I ...
3 votes
0 answers
56 views

### Intersection Form from a Kirby Diagram

If there are only 2 handles and no 1 handle in a Kirby Diagram then the intersection form of the underlying simply-connected 4-manifold coincides with the linking form. But what if there’s at least ...
1 vote
0 answers
57 views

• 183
0 votes
2 answers
239 views

### Who invented special framed links as a way to specify four-manifolds?

It is possible to specify (smooth, oriented, compact) four-manifolds with special framed links: $$I \times \mathbb{R}P^3$$ Such a link is divided in two sublinks, one ordinary link representing the 2-...
• 2,692
3 votes
2 answers
712 views

### Prerequisites for Kirby Calculus?

I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus? I have a solid ...
• 405
2 votes
1 answer
138 views

### How can uncountably many closed smooth 4-manifolds be presented by an essentially countable alphabet (Kirby diagrams)?

A smooth, closed 4-manifold admits a handle decomposition which is specified completely by its Kirby diagram. A Kirby diagram, up to isotopy, can be seen as a labelled morphism in the tangle category. ...
• 2,692
2 votes
1 answer
334 views

### How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
• 6,093
3 votes
0 answers
162 views

### How to get a Kirby diagram of $S^1 \times M^3$ if $M^3$ is given by a surgery diagram?

In "4-manifolds and Kirby Calculus" by Gompf and Stipsicz, there is a nice description of how to get the Kirby diagram of $S^1 \times M^3$, given a Kirby diagram of $M^3$. Basically, one thickens the ...
• 2,692
4 votes
1 answer
516 views

### Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
• 2,692
2 votes
0 answers
250 views

### Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
• 2,701
1 vote
1 answer
121 views

### Are there any combinatorial studies of Kirby calculus?

All of the other diagrammatic calculi I know of can be utilised with basically just combinatorial knowledge - for instance calculating knot and link polynomials. Are there similar combinatorial ...
5 votes
2 answers
688 views

### Cancelling 3-handles in Kirby diagrams

Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 ...
4 votes
1 answer
918 views

### About Kirby Diagrams

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 ...
• 227
3 votes
1 answer
122 views

### Kirby-like diagrams for $M^n$ when $n > 4$

Are there any attempts on constructing Kirby-like diagrams for representing manifolds $M^n$ with $n > 4$. What are the references on that ? I think you run out of dimension in which you can draw ...
• 2,701
4 votes
1 answer
569 views

### 4-Manifolds of which there exist no Kirby diagrams

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin ...
• 2,701