The 4-manifolds tag has no wiki summary.
3
votes
1answer
86 views
0-homologous surface bounds
Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented ...
10
votes
0answers
149 views
Fox re-imbedding theorem in dimension four
Fox re-imbedding theorem states the following:
A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
3
votes
1answer
212 views
What are Kirby diagrams of candidate exotic 4-manifolds?
It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle ...
4
votes
2answers
198 views
Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?
It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...
3
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0answers
116 views
Compact self-dual Einstein metrics of negative scalar curvature
Does a compact four-dimensional self-dual Einstein manifold with negative scalar curvature have negative sectional curvature?
This would be true if we believe the folklore conjecture that a compact ...
6
votes
2answers
221 views
Knotted projective planes and fake complex projective space
Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is ...
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0answers
99 views
K3 surface minus finite set
Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that
$$
H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z})
$$
for $0\le i \le 2$. Then the Euler ...
4
votes
1answer
233 views
Casson invariant and signature
In W. Neumann, J. Wahl, "Casson invariant of links of singularities",
Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...
3
votes
0answers
192 views
Is there a classification of closed self-dual 4-manifolds with positive sectional curvature?
In a paper of LeBrun-Nayatani-Nitta, they proved that a compact self-dual 4-manifold with positive Ricci curvature is homeomorphic to $n\,\mathbf{CP}^2$ for some $n \geq 0$, and diffeomorphic to ...
1
vote
0answers
132 views
complex Morse function on a four-manifold
If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under ...
0
votes
1answer
325 views
Stein manifolds definiton
There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex ...
0
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0answers
111 views
Affine varieties as Stein surfaces
I have a somewhat general and vague question in mind. Is there anything in literature related to Affine varieties as examples of Stein manifolds? I know that there is a topological approach to Stein ...
0
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0answers
200 views
What is topological surgery conjecture?
I'm a novice in the surgery theory, and I'm encountering the word "(4-dimensional) topological surgery conjecture", which I can't not find the definition. Could anyone help me?
2
votes
1answer
184 views
Gluck twist on four-manifolds
I have a basic question which I am not able to figure out. If we do a Gluck twist on a nullhomologous 2-sphere in a 4-manifold, it is said that it does not change its intersection form. But as far as ...
3
votes
1answer
289 views
Boundaries of smooth manifolds
If one has a smooth simply connected manifold $M^n$ which we know to bound an $n+1$ dimensional manifold $N$, what can be said about a handle decomposition for one in terms of a handle decomposition ...
14
votes
3answers
770 views
open problems in Seiberg-Witten Theory on 4-Manifolds
What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...
6
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0answers
182 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 ...
6
votes
1answer
286 views
Almost parallelizable 4-manifolds
On the first page of Milnor-Kervaire's paper "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", they assert without proof or reference that if $M$ is a compact connected oriented ...
10
votes
1answer
240 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 ...
17
votes
0answers
503 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 ...
4
votes
1answer
398 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 ...
9
votes
4answers
1k views
How Many 4-Manifolds are Symplectic?
As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...
3
votes
1answer
257 views
Intersection form on quotient manifold
Let $E_{1},E_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota_{i}$ the translation by a 2-torsion point on $E_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the the product ...
7
votes
1answer
571 views
A remark by Gromov on 4-manifolds
Gromov remarks in a a survey on manifolds (p.12) that "it is hard to imagine that there are infinitely many non-diffeomorphic, but mutually homeomorphic, quotients of the hyperbolic 4-space by ...
10
votes
1answer
1k views
What does Yang-Mills and mass gap problem has to do with mathematics?
I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and ...
1
vote
1answer
458 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, ...
4
votes
2answers
439 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 ...
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0answers
204 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 ...
1
vote
2answers
236 views
Group action on spin^c 4-manifold.
[edit]
I'll try to be more precise.
In paper N.Nakamura, "Bauer–Furuta invariants under $Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means:
...
2
votes
1answer
505 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) ...
6
votes
0answers
131 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 ...
10
votes
1answer
384 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 ...
8
votes
2answers
645 views
When is the connected sum of manifolds orientation-independent?
Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed?
If $N$ ...
5
votes
5answers
974 views
What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?
My question is as stated in the title:
What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?
I am particularly interested to know if ...
9
votes
1answer
347 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 ...
11
votes
2answers
445 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 ...
10
votes
1answer
477 views
Smooth four-manifolds with contractible universal cover
Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible?
It semms to me that the answer is negative when $X$ is simply ...
4
votes
3answers
844 views
Minimal genus, adjunction inequality
Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know ...
16
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0answers
721 views
When is a compact topological 4-manifold a CW complex?
Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...
10
votes
1answer
981 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 $CP^2$?
10
votes
1answer
565 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.
...
10
votes
2answers
382 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 ...
12
votes
2answers
503 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 ...
25
votes
0answers
886 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.
...
5
votes
2answers
517 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 ...
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0answers
241 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 ...
12
votes
1answer
896 views
A question on classification of almost complex structures on $4$-manifolds
I have a (basic?) question in topology.
Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by ...
23
votes
1answer
1k 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 ...
61
votes
4answers
5k views
How can there be topological 4-manifolds with no differentiable structure?
This is a very naive question, and I'm hoping that it will be matched by a correspondingly elementary answer. It is well known that not every topological 4-manifold admits a smooth structure. So ...
16
votes
1answer
783 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 ...
