# 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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### 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 $M$, ...

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### 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 ...

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### 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 $\omega$ ...

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### 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 ...

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### 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 \...

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### 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,...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 $p$-...

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### 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 $...

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### 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.

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### 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 ...

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### 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 ...

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### 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 $Y$...

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### Curvature of the line bundle $\mathcal{O}(2)$ on the twistor space

Let $M^4$ be a closed Riemannian manifold and $Z:=S\big(\Lambda^2_+(M)\big)$ denote the twistor space of $M,$ i.e., the sphere bundle of the self-dual 2-forms on $M$. Now at a point $(m,J)\in Z$ the ...

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### Are there connected closed 4-manifolds admitting a regular Almost Lagrangian distribution, and which are not Lorentzian?

In the category of real differential manifolds, connected (of $ C ^ {\infty} $ class in the sequel), closed of dimension 4, is there any manifold admitting a regular Almost Lagrangian distribution and ...

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### Submanifolds of $4$-manifolds and their intersections

Suppose we have two (oriented) submanifolds $A,B$ of an oriented $4$-dimensional manifold $M$, that intersect transversally. Looking at the standard references for $4$-manifolds, I couldn't find a ...

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### Presentations of exotic 4-manifolds

TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed).
Are there known presentations of $4$-manifolds $M$ with exotic structures, ...

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### A circle action on the E8 manifold

In the paper "A survey of group actions on 4-manifolds" by
Allan L. Edmonds on page 5 there is the remark "One should note that the coned-off E8-plumbing manifold
admits a circle action ...

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### Bound of the spinor element in Seiberg-Witten equation for a Kähler surface

Let's say we want to solve a perturbed version of SW equations on a closed Kähler manifold $(X,\omega):$
\begin{align*}
&D_A\phi=0\\
&F_A+it\omega=q(\phi)=\phi\otimes\phi^*-\frac{|\phi|^2}{2}\...

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### An Akbulut cork with a simple equation?

Is there an Akbulut cork that is diffeomorphic to a complex affine surface that can be given by a simple equation? (for example a surface given by zeros of a low-degree polynomial in $\mathbb C^3$)
...

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### Understanding the quadratic part in Seiberg Witten equation

Lets take a closed four manifold $M:=\Sigma_1\times \Sigma_2,$ where $\Sigma_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma_1$ and $\Sigma_2$ ...

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### A generalized Dirac operator

Let $(M^4,g)$ be a closed four-dimensional Riemannian manifold and $J$ be an almost complex structure on $M$. Then for normal coordinate $e_1,\dots e_4$ at a point $m,$ and for a section $\alpha$ of a ...

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### Configuration space of 4 points as an orbifold

Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...

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### h-cobordisms between non-simply-connected 4-manifolds

Let $M_0^4$ and $M_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M_0$ and $M_1$ and with ...

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### Contractible four-manifold which admits a decomposition

Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is ...

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### Characteristic disks in $S^2 \times S^2$ for knots

I'm studying the article Genera and degrees of Torus Knots in $\mathbb{CP}^2$ and I ended up with a question.
We know that every knot is slice (i.e. bounds properly embedded smooth disk) in $S^2 \...

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### Kahler surface with certain topology

Let $P^2 \mathrel{\tilde \times} \mathbb{R}^2$ be the $\mathbb{Z}_2$-quotient of $S^2 \times \mathbb{R}^2$, where the $\mathbb{Z}_2$ action on $S^2 \times \mathbb{R}^2$ is antipodal on $S^2$ and a ...

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### Contractible 4-manifold with an exhaustion

Let $M^4$ be a noncompact, contractible, 4-dimensional, topological manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) ...

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### Surgery diagrams of 3-dim abstract open books

Starting with an abstract open book $(\Sigma,\phi)$, I would like to understand some of the manifolds that I could obtain. Given a surface $\Sigma$ and monodromy $\phi$, it is not hard to find a Kirby ...

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### Generators of the automorphism group of a quadratic form

Suppose that $q$ is an integral quadratic form, not necessarily unimodular or even non-degenerate, and write $O_q(\mathbb{Z})$ for its automorphism group. According to this question this group is ...

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### Invertible 2-knots in $S^4$

Q1: Is it true that a knot $S^2\hookrightarrow S^4$ has an inverse iff it is trivial? Or it is also an open question?
See relatedly Unknotted $S^{n-2}$ in $S^n$.
Q2: It is easy to see that if a knot $...

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### Unknotted $S^{n-2}$ in $S^n$

I wonder is it still an open question that a smooth sphere $\Sigma^{2}\subset S^4$ is unknotted in $S^4$ iff its complement is homotopy equivalent to $S^1$? If it is an open question, how is it ...

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### Classification of genus 1 Lefschetz fibration over a disk with 6 singular fibers

For a genus 1 Lefschetz fibration over a sphere, there is a good classification in B. Moishezon, Complex surfaces and connected sums of complex projective planes. In this case, the number $K$ of ...

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### Concordance classes of diffeomorphisms of $D^4$

Is anything known about the set of concordance classes (also called pseudoisotopy classes) of the relative to the boundary diffeomorphisms of $D^4$?

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### Do different decompositions of Dehn twists induce the same cobordisms between mapping tori?

Suppose $Y=T^2\times I/f$ is a mapping torus, where $f$ is a diffeomorphism of $T^2$. Suppose $c$ is a curve on $T^2\times \{1\}$ with surface framing and suppose $D_c$ is the positive Dehn twist ...

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### The Kirby diagram of a manifold glued along the lens space $L(p,1)$

Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...

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### Questions about a paper by Laudenbach and Poénaru

I am working on the 1972 paper A Note on 4-Dimensional Handlebodies by F. Laudenbach and V. Poénaru, and I had two questions. I will use their notations to simplify things, since the paper is very ...

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### Involutions on $D^4$ with a fixed arc

By a theorem of Livesay, the 3-sphere has a unique (up to equivariant diffeomorphism) smooth involution with exactly two fixed points. Thinking of $S^3$ as the unit sphere in $\mathbb{R}^4$, this ...

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### Can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable, PL or DIFF manifold, if $X_4$ is a non-triangulable manifold? [duplicate]

Question: If $X_4$ is a non-triangulable topological (TOP) manifold,
can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable manifold?
can $X_4 \times S^1$, $X_4 \...

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### Characteristic class that cannot be represented by disjoint tori

Is there a simply-connected smooth closed 4-manifold with a characteristic class $x \in H_2(X; \mathbb{Z})$ such that $x$ can not be represented by a disjoint union of tori in $X$?
I would not know ...

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### Minimal genus of characteristic surfaces?

Let $X^4$ be a simply-connected closed smooth 4-manifold. Then every element $x \in H_2(X; \mathbb{Z})$ can be represented by an embedded orientable surface and the minimal genus of such a surface is ...

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### Mazur homology spheres

This is Example 6.47 in Saveliev's book Invariants for homology $3$-spheres:
Let us consider a two-component link $\mathcal L = L_1 \cup L_2$ in
$S^3$ such that $\mathrm{lk}(L_1,L_2) = \pm 1$ and the ...

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### Exotic $\mathbb{R}^4$ with a complex structure?

Is there an exotic $\mathbb{R}^4$ admitting an integrable almost complex structure?

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### $0$-surgery of slice knots and contractible manifolds

We know that if we attach $4$-dimensional $2$-handle $D^2 \times D^2$ to $S^1 \times S^2$, then we produce a contractible $4$-manifold. In this case, $S^1 \times S^2$ is $0$-surgery on the unknot.
If ...

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### Atiyah's proof of the moduli space of SD irreducible YM connections

In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a ...

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### Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?

Recall that two 4-manifolds $M$ and $N$ are stably diffeomorphic if there exist $m,n$ such that
$$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$
That is, they become diffeomorphic after ...

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### On the smoothness of plumbed $4$-manifolds and $E_8$ manifold

The $E_8$ manifold can be constructed by plumbing together disk bundles of Euler number $2$ over $2$-sphere $S^2$ according to the Dynkin diagram for $E_8$.
In Smooth 4-manifolds with $E_8$ ...

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### Very particular kind of 4-manifolds. Classification

Let $M$ be a smooth orientable compact connected (with boundary) manifold of dimension $4$. In addition $M$ is assumed to be aspherical and acyclic.
Question: is there a "classification" of ...