All Questions
13 questions
24
votes
0
answers
1k
views
Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
- 1,629
21
votes
0
answers
344
views
Are there exotic $\mathbb{R}^4$'s that are products with $\mathbb{R}$? [duplicate]
This came up in a conversation:
Question: Is there an exotic $\mathbb{R}^4$ that smoothly splits off an $\mathbb{R}$ factor? More precisely, suppose that $\mathcal R$ is a smooth 4-manifold which ...
- 724
16
votes
1
answer
569
views
Shortest Casson tower containing a slice disk for the attaching curve
A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk (...
- 711
11
votes
2
answers
844
views
Existing proofs of Rokhlin's theorem for PL manifolds
I'm looking for a comprehensive reference to existing proofs of Rokhlin's theorem that a 4-dimensional closed spin PL manifold has signature divisible by 16.
I'm specifically interested in direct ...
- 7,842
10
votes
2
answers
544
views
Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$
In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...
- 755
9
votes
1
answer
296
views
The works of González-Acuña and Duchon from 70s and 80s
I would like to access the following two works of González-Acuña from 1970:
González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79.
and
González-Acuña, F. On ...
- 1,292
8
votes
1
answer
2k
views
Relation of SW and Donaldson Invariant
My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space ...
- 703
7
votes
2
answers
366
views
Boundary of a $4$-manifold and the fundamental group
I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$,
Assume $\pi_1(N)$ is known,...
- 1,177
7
votes
0
answers
1k
views
Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...
- 10.5k
6
votes
0
answers
217
views
What is the state of the art in 4-manfold 2-types?
In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
- 35.5k
5
votes
2
answers
1k
views
First appearance of Novikov's additivity theorem
Hi!
Novikov's additivity theorem states that if you glue together two compact oriented 4n-manifolds along a connected component of their boundaries, the signature of the resulting manifold is ...
- 558
3
votes
1
answer
331
views
Simply connected 4-manifolds with boundary
I think I've encountered a question about 4-manifolds which maybe easy but I'm not familiar with. Can anyone give me an example of a simply connected 4-manifold $M$ (with boundary, of course) with $...
- 123
3
votes
1
answer
212
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 ...
- 213