# 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.

194
questions

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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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247 views

### 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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125 views

### 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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201 views

### 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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**1**answer

261 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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176 views

### $0$-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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827 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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381 views

### 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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110 views

### 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 ...

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693 views

### Homotopy groups of Diff(X) and Homeo(X)

For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general ...

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232 views

### Mazur and contractible manifolds

A Mazur manifold is a contractible, compact, smooth $4$-manifold with boundary a homology $3$-sphere.
It is built from a single $0$-handle, a single $1$-handle and single $2$-handle. It is equivalent ...

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475 views

### 4-dimensional cohomology $\mathbb{CP}^2$'s

Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?

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336 views

### Akbulut's cork involution

Akbulut's cork is the Mazur manifold $W$ shown in the picture below,
This manifold carries an involution of it's boundary $f:\partial W\to \partial W$ that exchanges a meridian of the 0-framed curve ...

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99 views

### May this slice disk for the unknot be pushed into the boundary?

Write the 4-ball as $\mathbb{D}^4=\mathbb{D}^2\times \mathbb{D}^2$.
Then its boundary $\mathbb{S}^3\simeq \mathbb{S}^1\times \mathbb{D}^2\cup \mathbb{D}^2\times \mathbb{S}^1$. We will use implicitely ...

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281 views

### Rational slice knot that is not slice

Does there exists a knot $K\subset \mathbb{S}^3$ such that
$K$ is not slice
$\exists W^4$, $\partial W = \mathbb{S}^3$ rational homology ball
$\exists $ properly embedded smooth disk $(D,\partial D)\...

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204 views

### Integer surgeries along links yielding lens spaces

Does there exist an integer $N$ such that any lens space $L(p,q)$ can be obtained by integer surgery from $S^3$ along a link $L$ with at most $N$ components?
EDIT:
I have worked out the comment by ...

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156 views

### Codimension two foliations with transverse surfaces

Suppose I have some closed $4$-manifold $X$ and a codimension-two foliation $\mathcal{F}$, as well as a closed surface $\Sigma$ of nonnegative self-intersection that is everywhere transverse to $\...

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### Status of a conjecture of C.T.C. Wall?

In Wall's paper Unknotting tori in codimension one and spheres in codimension two, he states the following conjecture:
Any $h$-cobordism of $S^3 \times S^1$ to itself is diffeomorphic to $S^3 \times ...

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183 views

### Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2

Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...

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### Is a gluing of homeomorphic Mazur manifolds diffeomorphic to $S^4$?

A recent paper proves the existence of homeomorphic but not diffeomorphic Mazur manifolds (see also examples of exotic pairs of contractible Stein manifolds).
Let's call them $M_1$ and $M_2$. If we ...

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259 views

### Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it

I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is ...

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186 views

### Smoothability of open 4-manifolds

F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5.
He first proves the generalized annulus conjecture:
Suppose $h:D^j\times \...

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209 views

### 3-balls with the same boundary in $S^4$ differ up to diffeomorphism

I am looking at this recent paper by Budney and Gabai and I am confused by a certain sentence in it. Theorem 4.7 states that if $\Delta_1$ and $\Delta_2$ are two 3-balls smoothly embedded in $S^4$ ...

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### Index of Dirac operator and Chern character of symmetric product twisting bundle

I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...

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### Self diffeomorphism of $S^2\times S^2$

The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2
Is it true that every self ...

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345 views

### Topological mapping class groups of 4-manifolds

It is a classical result of Quinn that for a simply-connected
closed $4$-manifold $X$ the isometries of its intersection form
are in one-to-one correspondence with
$\pi_0 \text{Homeo}(X)$. (Isotopy ...

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295 views

### Gauss-Bonnet-Chern Theorem [closed]

I am currently doing an undergraduate project about Gauss-Bonnet-Chern Theorem. Is there any particular books/papers regarding the application of the theorem in the theory of general relativity?

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### Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)

I'm studying Stefan Bauer's paper
The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...

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281 views

### Genus 2 3-manifolds bounding only $X^4$ with $b_2(X^4)$ big?

The genus of a closed orientable 3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$. Every such 3-manifold bounds a compact 4-manifold. Let $I(M)$ denote the ...

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### A 4-manifold with a special non-free circle action?

Let $X$ be an oriented closed 4 manifold, with a nontrivial orientation-preserving circle action.
Question Is there an example such that $X/S^1$ is an orbifold (not a manifold), with a trivial first ...

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### The effect of the Hodge $\star$ operator on the symplectic structure of a Kahler $4$ manifold

Let $(M,\omega, J, g)$ be a $4$ dimensional Kahler manifold. Put $\omega'=\star \omega$ where $\star$ is the Hodge operator associated the metric $g$.
Is $(M,\omega ')$ a symplectic manifold? Is it ...

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### $\exists X$?(No basis for $H_2(X)$ contains spheres)

Does there exist a smooth simply connected closed 4-manifold $X$ with the property below?
Every smooth basis for $H_2(X)$ contains a surface with genus $\geq 1$.
I understand that in general the ...

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### (Non-)Orientability of non-triangulable manifolds

We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...

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### Flat scalar curvature on 4 manifold

Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation.
Q Is $T^4$ the ...

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### ASD connection for Line bundle over $4$-manifold

Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...

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### Self-dual differential on $4$-manifold with boundary

Let $(M,g)$ be an oriented compact Riemannian $4$-manifold with boundary $\partial M$.
Let $a\in \Omega^1$ such that $*a\big|_{\partial M}=0$, i.e. $a(\nu)=0$ on $\partial M$, where $\nu$ denotes the ...

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### Lusternik-Schnirelmann Category of 4-Manifolds

Whilst reading 'the book' I stumbled across the following elegant little theorem, due to Gómez-Larrañaga and González-Acuña.
Let $M$ be a closed $3$-dimensional manifold. Then its Lusternik-...

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### Can an exotic diffeomorphism of the 4-ball change the isotopy class of an embedded surface?

Let $W$ be $B^4$ or $S^3 \times I$. Let $Y$ be a properly embedded surface in $W$. Let $f : W \to W$ be a diffeomorphism which is the identity near $\partial W$. Very little is known about $\pi_0(\...

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### Obtaining the bounding 4-manifold from the Heegaard diagram

It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...

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### Example of two exotic closed 4-manifolds s.t. SW(X)=0

I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which ...

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### Are triangulations with common refinements PL-homeomorphic?

Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...

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256 views

### A user guide to the theory on Corks

I am trying to digest the meanings of the corks from the both:
algebraic topology
and
geometry topology
perspectives.
Studying corks is important for understanding the exotic phenomenon of 4-...

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### Handlesliding a two component, linking number 1 link

Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...

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254 views

### Are there exotic twisted doubles of 4-manifolds?

Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...

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### Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?

Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...

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### Relation between the Casson-Gordon invariants $\sigma(M, \chi)$ and $\sigma_r(M, \chi)$

Setting: There are two objects in knot theory that are commonly referred to as the Casson-Gordon invariants: the invariant $\sigma$, and the invariant $\tau$ (see for example A. Conway’s notes ...