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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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
  • 51
3 votes
1 answer
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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
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2 votes
0 answers
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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
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3 votes
1 answer
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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
  • 8,417
10 votes
3 answers
491 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
  • 905
7 votes
2 answers
234 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
  • 905
2 votes
1 answer
141 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
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2 votes
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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
  • 131
6 votes
1 answer
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"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
99 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
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236 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
183 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
213 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
201 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
86 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
  • 525
5 votes
1 answer
250 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
349 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
134 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
76 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
  • 1,235
5 votes
1 answer
143 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
189 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
  • 1,019
2 votes
1 answer
302 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
  • 1,019
4 votes
1 answer
95 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
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8 votes
1 answer
166 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
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4 votes
0 answers
104 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
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7 votes
2 answers
337 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
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7 votes
1 answer
216 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
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2 votes
0 answers
83 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
396 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
74 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
7 votes
0 answers
204 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 around nineteen-seventies: González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79. and ...
Oğuz Şavk's user avatar
  • 1,242
2 votes
0 answers
88 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
  • 2,769
1 vote
0 answers
110 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
  • 1,019
2 votes
0 answers
169 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
  • 575
7 votes
2 answers
276 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,...
piper1967's user avatar
  • 1,019
5 votes
1 answer
253 views

Can we prescribe the $L^2$ norm of the scalar curvature on a four-manifold?

As mentioned by Willie Wong, I modified to the following version: Let $M$ be a closed smooth $4$ manifold. Q Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, ...
DLIN's user avatar
  • 1,895
7 votes
0 answers
232 views

On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
Anubhav Mukherjee's user avatar
12 votes
1 answer
371 views

Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?

This is a crosspost from this MSE question from a year ago. Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form? If $\omega$ ...
Michael Albanese's user avatar
8 votes
0 answers
256 views

A conjecture about homotopy $S^1\times B^3$'s

$\textbf{Conjecture}:$ Let $X^4$ be a homotopy $S^1\times B^3$ with the following properties: Attaching a four dimensional 2-handle gives a standard $B^4$. The $k$-fold cyclic cover is diffeomorphic ...
Daniel H. Hartman's user avatar
9 votes
2 answers
426 views

A knot in the solid torus and a Mazur manifold

Part 1: The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130: He indicates that the knot drawn in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \...
Terry Black's user avatar
5 votes
1 answer
600 views

Kirby diagrams: sliding 1-handles over 1-handles and ribbon disks

Consider the Kirby diagram $ D$ given by a 2-component unlink, both dotted circles. In general, when performing a 1-handle slide over another 1-handle, the band chosen must not link any dotted circle,...
Overflowian's user avatar
  • 2,473
8 votes
2 answers
254 views

Gordon's approach: slice knots and contractible $4$-manifolds

Let $K \subset S^3$ be a slice knot. Then it bounds a smooth embedded disk $D \subset B^4$. Let $S^3_{p/q}(K)$ denote a $3$-manifold obtained by $p/q$-surgery on $K \subset S^3$. The following theorem ...
Max Schumann's user avatar
6 votes
1 answer
271 views

Computation of $\pi_1$ for a Mazur manifold and its boundary

If we attach a $4$-dimensional $1$-handle $D^1 \times D^3$ to a $4$-dimensional $0$-handle $B^4$, we obtain $S^1 \times D^3$. The null homologous knot in $S^1 \times S^2$ indicated in the picture ...
Max Schumann's user avatar
6 votes
2 answers
433 views

Status of the Hopf-Thurston sign conjecture in dimension 4

A famous conjecture in topology asserts: The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$. This was conjectured by Hopf for manifolds with non-...
Jens Reinhold's user avatar
6 votes
2 answers
248 views

Double ($p$-fold) coverings of $B^4$ along ribbon/slice disks

I have two questions that seem to be related. I wonder if there is a user-friendly algorithm (starting from ribbon/slice presentation of knots/disks) for the construction of double (in general $p$-...
Max Schumann's user avatar
2 votes
0 answers
173 views

plumbing description of resolution of ADE singularities

Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $...
user44651's user avatar
  • 989
2 votes
0 answers
119 views

Seiberg Witten to Vortex equations

Looking for a reference which discusses vortex equations on a compact Riemann surface as dimensional reductions of the Seiberg-Witten equations on four manifolds.
Partha's user avatar
  • 595
9 votes
1 answer
364 views

Non-isotopic homology spheres in $S^4$ with equal complements?

Are there two diffeomorphic smoothly embedded homology 3-spheres $M_1^3, M_2^3 \subset S^4$ that have diffeomorphic complements but such that $M_1$ and $M_2$ are not isotopic? I would be interested in ...
user101010's user avatar
  • 5,299
4 votes
0 answers
314 views

Obstruction of smooth structure

The first 24 lectures of Jacob Lurie on Geometric Topology [1] gave a concise introduction to the comparison of smooth manifolds and piecewise-linear manifold. In the first five lectures, it is shown ...
Student's user avatar
  • 4,560
10 votes
0 answers
314 views

Isotopy on embedded 3-manifolds in 4-manifolds

Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
Anubhav Mukherjee's user avatar

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