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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.

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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 $\partial N=M$, ...
Zhiqiang's user avatar
  • 881
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}=...
Boyu Zhang's user avatar
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?
Zhiqiang's user avatar
  • 881
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 ...
Judy_xyh's user avatar
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 ...
Alessio Di Prisa's user avatar
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 $Y\times\{1\}$, $K$...
Qiuyu Ren's user avatar
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 \...
Zhiqiang's user avatar
  • 881
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 ...
Zhiqiang's user avatar
  • 881
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) ...
Moishe Kohan's user avatar
  • 9,817
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 ...
rab's user avatar
  • 159
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 $S^3$...
rab's user avatar
  • 159
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 $...
Upside Down's user avatar
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 $S^{3}$ ...
Elliot's user avatar
  • 295
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(...
rab's user avatar
  • 159
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 ...
Piotr Hajlasz's user avatar
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 ...
math-physicist's user avatar
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 ...
zeta's user avatar
  • 427
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 ...
ThorbenK's user avatar
  • 1,185
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,
maden's user avatar
  • 41
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-...
Zhaoting Wei's user avatar
  • 8,727
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. ...
ThorbenK's user avatar
  • 1,185
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 ...
ThorbenK's user avatar
  • 1,185
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 ...
piper1967's user avatar
  • 1,059
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{...
blancket's user avatar
  • 171
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 ...
Audrey Rosevear's user avatar
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 ...
asymmetriad's user avatar
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 ...
Shaoyun Bai's user avatar
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 $b_2(W)=1$? ($\cdots$...
user302934's user avatar
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 ...
Max Schumann's user avatar
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 ...
Terry Black's user avatar
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) &...
Faniel's user avatar
  • 653
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?
Anton Petrunin's user avatar
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 ...
user302934's user avatar
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;\...
user302934's user avatar
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 ...
Adterram's user avatar
  • 1,361
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 ...
user302934's user avatar
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 ...
piper1967's user avatar
  • 1,059
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 \...
piper1967's user avatar
  • 1,059
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 ...
M. Winter's user avatar
  • 12.7k
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 \...
M. Winter's user avatar
  • 12.7k
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 ...
Shijie Gu's user avatar
  • 2,043
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 ...
J. GE's user avatar
  • 2,593
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 ...
Shijie Gu's user avatar
  • 2,043
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 $M$...
John Samples's user avatar
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 ...
user302934's user avatar
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 ...
user302934's user avatar
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 ...
Oğuz Şavk's user avatar
  • 1,292
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 $x$ ...
Andi Bauer's user avatar
  • 2,911
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 ...
piper1967's user avatar
  • 1,059
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 ...
tota's user avatar
  • 585

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