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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 $Q$ ...
Hrhm's user avatar
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2 votes
1 answer
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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 ...
Prerak Deep's user avatar
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 ...
horned-sphere's user avatar
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 ...
horned-sphere's user avatar
1 vote
0 answers
57 views

$E(n)\# \mathbb{CP}^2=2n\mathbb{CP}^2\# (10n-1)\overline{\mathbb{CP}^2}$

How does one prove the following statement : $$E(n)\# \mathbb{CP}^2=2n\mathbb{CP}^2\# (10n-1)\overline{\mathbb{CP}^2}$$ Here $E(n)$ denotes the $n$-th elliptic surface formed by fiber summing $E(1)$ $...
Chanel Rose's user avatar
2 votes
0 answers
33 views

Can framings on plumbed manifolds be taken to be even?

The definition of "plumbed manifold" that I'm using in this context is the following - given a weighted tree $\Gamma$, build up a framed link $L(\Gamma)$ by chaining together two copies of ...
Perturbative's user avatar
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1 vote
1 answer
68 views

Picturing twisting of strands explicitly after blow downs

In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
Terry Black's user avatar
6 votes
3 answers
299 views

Way to determine the type of a knot given by a diagram

Is there a general way to determine the type of a knot given by a diagram? I am using KLO (Kirby calculator) and I encounter some nontrivial knots while doing this. For example, can we determine the ...
user302934's user avatar
  • 1,630
4 votes
1 answer
310 views

Are these two links equivalent?

Are the following two links equivalent (orientation preserving isotopies)? The two links have the same linking number. The only difference is the crossing that in one case is positive while in the ...
Overflowian's user avatar
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4 votes
0 answers
243 views

Not every 3-manifold is a graph manifold

Surgery presentations It is well known that any orientable closed 3-manifold $M$ has a surgery presentation, i.e. can be obtained by doing surgery on $\mathbb{S}^3$ on a link $L$. We can also ...
Overflowian's user avatar
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4 votes
1 answer
458 views

Kirby calculus on E8 plumbing

I was trying to show that the 4-manifold described in Kirby diagram as a E8-plumbing (see the diagram below) has the same boundary as the 2-handlebody on the left-handed trefoil with surgery ...
cjackal's user avatar
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3 votes
3 answers
431 views

Integer homology sphere as subsequent surgery on integer homology spheres

By Lickorish and Wallace , any closed,connected, orientable 3-manifold can be gotten as a surgery on a link in $S^3$. Let say our manifold, M, is an integer homology sphere and L = $ L_1 \cup L_2 \cup ...
Subhankar D.'s user avatar
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-...
Turion's user avatar
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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 ...
kingdras's user avatar
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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. ...
Turion's user avatar
  • 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 ...
Kyle's user avatar
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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 ...
Turion's user avatar
  • 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 ...
Turion's user avatar
  • 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 ...
Willem Noorduin's user avatar
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 ...
Gary Garygary's user avatar
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 ...
Rob Smith's user avatar
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 ...
T.O.'s user avatar
  • 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 ...
Willem Noorduin's user avatar
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 ...
Willem Noorduin's user avatar