# Questions tagged [4-manifolds]

A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.

282
questions

1
vote

0
answers

97
views

### Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.
Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...

15
votes

0
answers

289
views

### Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements

Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements?
And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...

3
votes

1
answer

168
views

### Principal circle bundles over punctured $3$-sphere

Let $M$ be $S^3$ with $k$ disjoint open balls $D^3$ removed.
Can we classify all principal circle bundles over $M$ such that the total space is simply-connected?

5
votes

1
answer

148
views

### Is there a nontrivial ribbon knot concordance from a knot to itself?

It was conjectured by Gordon and recently proved by Agol that ribbon concordance defined a partial order on the semi group of knots. I know that this question is close related to the slice ribbon ...

6
votes

1
answer

121
views

### Knotted concordances of slice links

Are there any examples of a link $L$ such that:
$L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap ...

6
votes

2
answers

345
views

### Slice knots in 3-manifolds

Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$...

10
votes

2
answers

1k
views

### Does the Hodge star operator determine the metric?

Let $M$ be a closed oriented smooth $4$-manifold with two metrics $g$ and $\tilde g$. Consider the Hodge star operators on $2$-forms
$$
\star:\Omega^2(M) \to \Omega^2(M) \quad \text{and} \quad \tilde \...

1
vote

0
answers

51
views

### Perturbation of the self-dual harmonic $2$-forms

Let $M$ be a closed oriented smooth $4$-manifold with $m=b^+_2(M)>0$. We set $\mathcal R$ to be the space of smooth metrics on $M$ and consider the following map $f:\mathcal R \to Gr(m,n)$ defined ...

14
votes

0
answers

333
views

### Nonsmoothable 4-manifolds

Does there exist a closed connected nonsmoothable 4-manifold $M$ such that:
$\kappa(M)=0$ (Kirby-Siebenmann invariant vanishes, hence, there is no "classical" obstruction to smoothability) ...

1
vote

0
answers

187
views

### Local to global complexity of triangulations

Alright 3rd time's the charm - editing again to put all my cards on the table.
Consider a PL $n$-manifold $M$. Define the complexity $c(M)$ of $M$ to be the minimum number of $n$-simplices needed to ...

4
votes

1
answer

433
views

### Detecting a "bad map" in Fintushel-Stern knot surgery

Background
Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...

5
votes

1
answer

291
views

### Kirby diagrams of Mazur manifolds

In the 1980's, Fintushel-Stern and Fickle independently proved that Brieskorn spheres $\Sigma(2,3,25)$ and $\Sigma(3,5,19)$ bound some Mazur type contractible 4-manifolds with a single $0$-, $1$, and $...

2
votes

0
answers

116
views

### Homotopy type of a 3-manifold produced via Dehn surgery?

My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.
I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...

6
votes

0
answers

152
views

### Kirby diagram of Enriques surface (as the "(1/2) K3 surface")

Not to be confused with $E(1)\cong\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, which is also known as a $\frac{1}{2}K3$ surface (in the sense that removing a neighbourhood of a regular torus fiber in $E(...

12
votes

0
answers

395
views

### Is the Lipschitz structure on $\mathbb{S}^4$ unique?

Sullivan [S] proved that on any $n$-dimensional, $n\neq 4$, topological manifold there is a unique Lipschitz structure. Although it is generally accepted result, it seems that the proof lacks some ...

3
votes

0
answers

468
views

### What are some of the big open problems in $4$-manifold theory?

I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...

4
votes

1
answer

398
views

### 4-manifold $M$ with intersection form of Leech lattice

Do we know any 4-manifold $M$ with intersection form of rank-24 Leech lattice?
Is $M$ smooth? Differentiable to which $n$-order?
Can $M$ be obtained by any surgery on 3 copies of E8 4-manifolds with ...

5
votes

1
answer

275
views

### 0-surgery on a fibered hyperbolic ribbon knot

Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered?
I tried looking at ...

2
votes

0
answers

95
views

### Chern number of positive spinor bundle

What is the second chern number $c_2(V_+)$ of the positive spinor bundle on a 4-manifold, in particular $S^4$? Why is it that $V_+$ is the same as the quaternion line-bundle?
Thanks,

3
votes

1
answer

166
views

### Do we have $d(P_{+}(\omega\wedge \theta))=d_{+}\omega\wedge \theta-\omega\wedge d_{+}\theta$ on a self-dual manifold?

Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of self-...

10
votes

3
answers

619
views

### Doubles of 2-handlebodies

Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. ...

8
votes

2
answers

343
views

### Open book decompositions in dimension 4

The question about the existence of open book decompositions for a closed oriented $n$-dimensional manifold seems to be answered in all dimensions except dimension $4$, where as far as I can tell the ...

2
votes

1
answer

160
views

### Homological restrictions on certain $4$-manifolds

I am not very familiar with the non-compact $4$-manifold theory. So I apologize if the following question is very silly.
Let $X$ be a non-compact, orientable $4$ manifold that is homotopic to an ...

2
votes

0
answers

78
views

### Arf invariant for characteristic surfaces in closed 4-manifolds depends on homeomorphism type

Let $X$ be a closed smooth oriented 4-manifold with $H_1(X;\Bbb Z)=0$. Then for a smoothly embedded orientable surface $F\subset X$ which is characteristic, there is a well-defined invariant $\text{...

6
votes

1
answer

300
views

### "Neck cutting" and why gauge theory doesn't work on homotopy 4-spheres

I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to ...

5
votes

0
answers

125
views

### Can every homeomorphism of the 4-ball fixing the boundary be approximated by diffeomorphisms smoothly isotopic to the identity?

Let $B^{n}$ denote the euclidean closed ball of dimension $n$. Alexander's trick shows that every homeomorphism of $B^{n}$ fixing $\partial B^n$ is $C^{0}$-isotopic to the identity via a boundary ...

6
votes

0
answers

253
views

### 11/8-type inequality from Heegaard Floer theory?

Background: a well-known conjecture in $4$-dimensional topology states that if $M$ is a $4$-dimensional oriented closed spin smooth manifold, then the second Betti number of $M$ is bounded from below ...

7
votes

1
answer

266
views

### Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$

The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf.
($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...

3
votes

1
answer

267
views

### A Mazur manifold bounded by $\Sigma(2,3,13)$

Using Kirby calculus, Akbulut and Kirby first analyzed that the Brieskorn sphere $\Sigma(2,3,13)$ is diffeomorphic to the following link in their famous paper:
Then they switched the circles when ...

3
votes

1
answer

208
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 ...

5
votes

0
answers

104
views

### Standard 2-instantons on the 4-sphere under conformal transformation

It is well-known that there is a standard SU(2) 1-instanton on the 4-sphere and any 1-instanton can be obtained from the standard one by conformal transformation. Is there any explicit (family of) &...

5
votes

1
answer

288
views

### Stable torus that is not a torus [duplicate]

Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?

6
votes

1
answer

380
views

### Two surfaces in a 4-manifold whose algebraic intersection number is zero

Suppose $X$ is a smooth closed oriented 4-manifold, and $\Sigma_1,\Sigma_2$ are smoothly embedded compact oriented surfaces in $X$. Suppose they intersect transversally at two points with different ...

4
votes

1
answer

239
views

### Homology classes in connected sum of $\Bbb CP^2$'s that can be represented by smoothly embedded spheres

Let $h=[\Bbb CP^1]\in H_2(\Bbb CP^2;\Bbb Z)$. By a theorem of Kronheimer and Mrowka (Theorem 1 of this paper: https://people.math.harvard.edu/~kronheim/thomconj.pdf), a class $nh \in H_2(\Bbb CP^2;\...

2
votes

0
answers

82
views

### Positivity of the intersection form [closed]

Let $M^4$ be a closed orientable smooth manifold and
$$
I: H_2(M,\mathbb Z) \times H_2(M,\mathbb Z) \to \mathbb Z
$$
its intersection form. If $b^+ \ge 1$, where $b^+$ denotes the number of positive ...

5
votes

1
answer

205
views

### Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?

In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3_{\Bbb Z}\to \Bbb Z$, where $\Theta^3_{\Bbb Z}$ is ...

2
votes

0
answers

194
views

### $4$-manifolds with boundary homotopic to $K(G,1)$

I am not very conversant with the $3$-manifold or $4$-manifold theory. The literature I found so far related to $3$-manifold tells me that $S^1\times S^2$ is "very special". It is prime but ...

2
votes

1
answer

344
views

### $4$-manifold with simply connected boundary

This may be a very silly question but I could not get any counter-example.
Let $M$ be a compact differential $4$-manifold with boundary $dM$.
Suppose that the inclusion map induced map $\pi_1(dM) \to \...

4
votes

1
answer

105
views

### A neighborhood of a 2-disc $D\subset\Bbb R^4$ that tapers off towards the boundary?

I am given a PL 2-disc $D\subset\Bbb R^4$ (everything PL from here on) and I need a "neighborhood" $N\simeq B^4$ (PL-homeomorphic to a 4-ball) so that $\partial N\cap D=\partial D$.
If I got ...

8
votes

1
answer

217
views

### If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?

Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.
Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \...

4
votes

0
answers

114
views

### Candidates of nonsmoothable homology 4-spheres

I'm not trying to duplicate a question Are there non-smoothable homotopy/homology spheres?
But in Igor Belegradek's answer, he pointed out that "Some of them (homology spheres) have large ...

7
votes

2
answers

411
views

### Examples of homology sphere that bound a nonsmoothable contractible 4-manifold

Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible ...

7
votes

1
answer

346
views

### Does Freedman's disk embedding theorem extend to infinitely many immersed disks?

I'm referring to a statement of the disk embedding theorem on P17 in a book of Behrens-Kalmar-Kim-Powell-Ray, the Disk Embedding Theorem. See below. There are some new formulations, e.g., Theorem 1.2 ...

2
votes

0
answers

92
views

### If a Compact $n$-Manifold Immerses in $\mathbb{R}^{n+1}$ is there a Locally Flat Immersion?

Suppose that $M$ is a compact, topological $n$-manifold and there is a topological immersion (i.e. local embedding) of $M$ into $\mathbb{R}^{n+1}$. Is there necessarily a locally flat immersion of $M$...

6
votes

1
answer

452
views

### 4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere

I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation:
Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...

3
votes

0
answers

84
views

### Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s

Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...

9
votes

1
answer

290
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 ...

2
votes

0
answers

117
views

### Attaching a 2-handle to a once-twisted unlink in the boundary of the 4-ball

Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ ...

1
vote

0
answers

128
views

### Middle Betti number bound of special $4$-manifolds

I was thinking of the following question regarding $4$-manifolds as follows.
Let $M$ be a compact, oriented, smooth $4$-manifold with a smooth connected boundary, say $N$. Let $M$ be simply connected ...

2
votes

0
answers

179
views

### When does a map between 4-manifolds map boundary to boundary upto homotopy?

Let $f:M\to N$ be a smooth map between smooth 4-manifolds with boundary. When does $f$ map boundary of $M$ to boundary of $N$ upto homotopy i.e. when there is a map $F:M\to N$ homotopic to $f$ such ...