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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What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
Bruno Martelli's user avatar
30 votes
1 answer
2k views

When is a compact topological 4-manifold a CW complex?

Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". Kirby showed that ...
Andrew Ranicki's user avatar
24 votes
0 answers
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Exotic 4-spheres and the Tate-Shafarevich Group

The title is a talk given by Sir M. Atiyah in a conference with the following abstract: I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
mathphys's user avatar
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24 votes
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Concordance and homology cobordism

If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if $...
Adam Levine's user avatar
19 votes
0 answers
839 views

Which manifolds decompose into pants?

In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a ...
Bruno Martelli's user avatar
18 votes
0 answers
868 views

Almost complex 4-manifolds with a "holomorphic" vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$? The following sub question is ...
Dmitri Panov's user avatar
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17 votes
0 answers
338 views

Are there two non-equivalent exotic structures on $\mathbb{R}^4$ coming from topologically slice, non-slice knots?

For a knot $K \subset S^3$, which is topologically slice but not slice (in a smooth way), there's a four manifold $\mathbb{R}^4_K$, homeomorphic but not diffeomorphic to standard euclidean $\mathbb{R}^...
Saman Habibi Esfahani's user avatar
15 votes
0 answers
271 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
14 votes
0 answers
316 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-...
Kyle Hayden's user avatar
13 votes
0 answers
321 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
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12 votes
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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
12 votes
0 answers
247 views

Excluding exotic PL structures on S^4

Suppose you have a finite group $G<SO(5)$ such that $S^4/G$ is homeomorphic to $S^4$ and such that $S^4/G$ is a PL manifold with respect to a PL structure induced by a standard structure on $S^4$. ...
Christian Lange's user avatar
11 votes
0 answers
349 views

Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following: A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
Bruno Martelli's user avatar
10 votes
0 answers
319 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
10 votes
0 answers
203 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,...
Julien Marché's user avatar
9 votes
0 answers
166 views

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= ...
Ethan Dlugie's user avatar
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8 votes
0 answers
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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 ...
Daniel H. Hartman's user avatar
8 votes
0 answers
403 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 ...
Marc Kegel's user avatar
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8 votes
0 answers
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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 ...
Anubhav Mukherjee's user avatar
8 votes
0 answers
296 views

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-...
Tyrone's user avatar
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7 votes
0 answers
286 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
7 votes
0 answers
1k views

Applications of E8 manifold

The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ ...
wonderich's user avatar
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7 votes
0 answers
495 views

Intersection form of logarithmic transformations

Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows. Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$ be an ...
Hopf Fibration's user avatar
6 votes
0 answers
144 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
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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
6 votes
0 answers
189 views

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 ...
Partha's user avatar
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6 votes
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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 ...
Anthony's user avatar
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6 votes
0 answers
179 views

Quotients of 4-sphere by smooth $Z_p$ actions with knotted fixed point sets

This question is closely related to another I asked today. Giffen showed in 1966 that the generalized Smith conjecture is false by constructing for odd $p$ a smooth $Z_p$ action on $S^4$ with fixed-...
benblumsmith's user avatar
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6 votes
0 answers
216 views

What is the state of the art in 4-manfold 2-types?

In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
David Roberts's user avatar
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6 votes
0 answers
196 views

Surgering locally flat tori in 4-manifolds

Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching ...
Daniele Zuddas's user avatar
5 votes
0 answers
119 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
5 votes
0 answers
100 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
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5 votes
0 answers
166 views

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 ...
Partha's user avatar
  • 759
5 votes
0 answers
165 views

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 ...
Eric Arnéo Vespira Kengne's user avatar
5 votes
0 answers
138 views

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 ...
Danny Ruberman's user avatar
5 votes
0 answers
302 views

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,...
user267839's user avatar
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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
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4 votes
0 answers
351 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
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4 votes
0 answers
146 views

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 ...
Nick A.'s user avatar
  • 203
4 votes
0 answers
225 views

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 $\mathbb C^3$) ...
aglearner's user avatar
  • 14k
4 votes
0 answers
114 views

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 ...
no_idea's user avatar
  • 459
4 votes
0 answers
123 views

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 ...
Strongly Negative Amphicheiral's user avatar
4 votes
0 answers
320 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 \...
cork_twist's user avatar
4 votes
0 answers
109 views

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 ...
user146582's user avatar
4 votes
0 answers
159 views

(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 ...
annie marie cœur's user avatar
4 votes
0 answers
244 views

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 ...
user136604's user avatar
4 votes
0 answers
150 views

Blow-up and Blow-down kirby local moves for non-orientable $3$-manifold

Can anyone explain or give a reference about the Blow-up and Blow-down Kirby local moves for non-orientable $3$-manifolds? Thanks, advance.
Selvakumar A's user avatar
4 votes
0 answers
217 views

Blow-up of $\mathbb{P}^4$ along a smooth surface

Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$). In dimension ...
Jérémy Blanc's user avatar
4 votes
0 answers
325 views

Compact self-dual Einstein metrics of negative scalar curvature

Does a compact four-dimensional self-dual Einstein manifold with negative scalar curvature have negative sectional curvature? This would be true if we believe the folklore conjecture that a compact ...
Shinichiroh Matsuo's user avatar
3 votes
0 answers
82 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