All Questions
Tagged with gt.geometric-topology 4-manifolds
179 questions
10
votes
1
answer
489
views
Visualising locally flat embeddings of surfaces in R^4
As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally ...
- 14.3k
19
votes
0
answers
852
views
Which manifolds decompose into pants?
In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a ...
- 10.5k
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
2
votes
1
answer
708
views
Problem:Gromov-Witten;Moduli space
Let us consider a map from a $\Sigma_g \longrightarrow N$, where $N$ is a symplectic manifold.
Then we define the moduli space as
$M= \{ f | f \mbox{ is a pseudoholomorphic map } \Sigma_g \to N, ...
- 703
5
votes
2
answers
1k
views
Intersection forms of 4-manifolds with boundary
Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
- 53
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
3
votes
1
answer
1k
views
Homeomorphism classification of 4-manifolds
Question 1. Let $X_i$ be an infinite family of closed, orientable, smooth 4-manifolds with the following properties:
a) $\pi_1(X_i) = \mathbb{Z}\times \mathbb{Z_{2}}$ for any $i = 1, 2, \cdots $
b) ...
- 45
6
votes
0
answers
199
views
Surgering locally flat tori in 4-manifolds
Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching ...
- 2,283
12
votes
1
answer
832
views
Handlebody decomposition of an open 4-manifold
Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth ...
- 2,283
9
votes
1
answer
627
views
Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively
I'm interested in a complexity question related to problems like the slice-ribbon problem.
To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...
- 44.4k
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
11
votes
1
answer
2k
views
Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]
Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?
- 129
15
votes
1
answer
1k
views
Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?
Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper.
Heegaard ...
- 724
10
votes
2
answers
585
views
Embedding the product of three circles in the 4-sphere.
Let $M$ be a smooth submanifold of the 4-sphere $S^4$. I'm going to demand that $M$ be diffeomorphic to $S^1 \times S^1 \times S^1$. By Jordan-Brouwer separation, $M$ separates the 4-sphere into ...
- 44.4k
13
votes
2
answers
1k
views
Explicit embeddings of Cappell-Shaneson knots
In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare ...
- 44.4k
37
votes
0
answers
2k
views
What is the three-dimensional hyperbolic volume of a four-manifold?
Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
...
- 10.5k
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
1
vote
0
answers
323
views
3-handle cancellation of 4-dimensional handlebody.
Let $X^4$ be the 4-dimensional handlebody with $\partial X=S^3$ and $\pi_i(X)=\pi_i(B^4)$.
Is it true that we can always change $X^4$ with handlebody without 3-handle?
(I'm concerning about the ...
- 11
28
votes
1
answer
2k
views
Can you flip the end of a large exotic $\mathbb{R}^4$
Can you flip the end of a large exotic $\mathbb{R}^4$
Background
Definition (Exotic $\mathbb{R}^4$):
An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to $\mathbb{...
- 1,897
19
votes
1
answer
1k
views
What's the Kirby Diagram of a universal $\mathbb{R}^4$?
What's the Kirby diagram of a universal $\mathbb{R}^4$?
Background
Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...
- 1,897
17
votes
1
answer
2k
views
Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)
A celebrated theorem of Rohlin states the following
An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero.
Simple homological arguments ...
- 10.5k
28
votes
4
answers
2k
views
Can all n-manifolds be obtained by gluing finitely many blocks?
Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
- 10.5k
19
votes
3
answers
2k
views
topological "milnor's conjecture" on torus knots.
Here's a question that has come up in a couple of talks that I have given recently.
The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly ...
- 653
10
votes
2
answers
910
views
slice=ribbon generalization to higher genus + potential counterexamples to slice=ribbon.
I have two questions about the slice=ribbon conjecture.
(1) If a knot $K \hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 \times [0, -\...
- 653
4
votes
4
answers
1k
views
4-genus of a 2-bridge link
How can we calculate the 4-genus of a link L? The 4-genus is defined to be the minimal genus of orientable surface bounded by L in B^4. Is there any routine method to calculate that?
Especially, any ...
- 1,040
0
votes
1
answer
415
views
If the 4-genus of a link is zero, is it a slice link?
An n-component slice link is a link that bounds n disjoint discs in B^4. And the 4-genus of a link is defined to be the minimal genus of orientable surfaces bounded by it in B^4.
My question is: if ...
- 1,040
18
votes
0
answers
873
views
Almost complex 4-manifolds with a "holomorphic" vector field
Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is ...
- 28.9k
29
votes
3
answers
2k
views
Embeddings of $S^2$ in $\mathbb{CP}^2$
Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect ...
- 6,247
8
votes
2
answers
1k
views
A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve?
There's an old problem in 4-manifold theory that, as far as I know, doesn't have a name associated with it and really deserves a name.
Let $M$ be a smooth 4-manifold with boundary. Let $S$ be a ...
- 44.4k