# 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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### 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 ...
281 views

### Obstruction of smooth structure

The first 24 lectures of Jacob Lurie on Geometric Topology  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, ...
131 views

### 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 $\mathbb C^3$) ...
251 views

### 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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### 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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### 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 ... 329 views

### 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 ... 864 views

### 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 ...
### 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 $E_8$ ... 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 ...