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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2
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0answers
101 views

$\exists X$?(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 ...
4
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0answers
95 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 6, ...
2
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1answer
130 views

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 ...
6
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1answer
143 views

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^+...
4
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1answer
145 views

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 ...
8
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252 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-...
6
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1answer
182 views

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(\...
3
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1answer
159 views

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 ...
11
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1answer
381 views

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 ...
4
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0answers
110 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 ...
3
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201 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-...
3
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74 views

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 ...
13
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0answers
213 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-...
13
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1answer
307 views

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 ...
8
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1answer
175 views

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 ...
5
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284 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 \\ ...
5
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168 views

Homology spheres bounding homology balls but not embedding into $S^4$

Are there any examples of integer homology spheres $Y^3$ that bound smooth integer homology balls but that do not smoothly embeded into $S^4$?
6
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1answer
252 views

smooth homotopy 4-balls with sphere boundary in dimension 4

What follows is, as far as I can tell, totally standard folklore. I have one particular point of confusion, other than that, I wanted to confirm that I am uttering the incantations correctly. The ...
6
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1answer
189 views

Framings for 2-surgeries on 4-manifolds

I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$. Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible ...
4
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1answer
130 views

Simple invariants to detect concordance in general 3-manifolds

Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that ...
2
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1answer
209 views

Embedding problem for 3-manifolds attacked via 4-manifolds

In this archiv paper which is continuation of following: Borodzik, Maciej; Némethi, András; Ranicki, Andrew, Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16, No. 2, 971-1023 (2016). ...
36
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1answer
719 views

Exotic $R^4$ as the universal covering space

Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$? Remark. I am aware of examples (due to Mike Davis) of compact $...
9
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1answer
459 views

Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?

Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and ...
7
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1answer
104 views

Minimum number of double points over all immersed disks

Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to ...
9
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3answers
956 views

Is there a closed non-smoothable 4-manifold with zero Euler characteristic?

I will just repeat the title: Is there a closed non-smoothable 4-manifold with zero Euler characteristic? I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
22
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1answer
661 views

What can we say about the Cartesian product of a manifold with its exotic copy?

Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$. Is it true that $M\times M$ is diffeomorphic to $M\times M^E$? I am ...
18
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2answers
534 views

Behavior of genus function on a 4-manifold for sums

Let $X$ be a smooth compact 4-manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \...
3
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0answers
83 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.
4
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1answer
275 views

Canonical Metrics on 3- and 4-Manifolds

From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
6
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1answer
452 views

What is “topology in dimension 3.5”?

I've noticed a couple of conference titles which reference something called "topology in dimension 3.5," such as this one and this one. This subject seems quite mysterious to me — it looks like ...
5
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2answers
832 views

Topological Classification of Four-Manifolds

It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. ...
34
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3answers
778 views

How to specify a compact topological 4-manifold with a finite amount of data

Finite data for 4-manifolds: I have a somewhat hazy memory of seeing a Ph.D. thesis around 2005 which showed that any compact topological 4-manifold $M$ can be specified by a finite amount of data. ...
4
votes
2answers
398 views

Are there Kirby diagrams with 3-handles?

Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\...
2
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1answer
133 views

Lefschetz Fibrations and disk bundles

When reading chaptes 7 of Akbulut's book about $4-$manifolds, he describes a handle decomposition for a manifold given a Lefschetz fibration over $S^2$. The idea is to extend the preimage of a disk ...
6
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1answer
301 views

Fundamental groups of non-orientable closed four-manifolds

The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see this question. ...
6
votes
2answers
337 views

Decribe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$

According to this MO post, there is two possible $S^2$ fibration over $S^2$. One is obviously $S^2\times S^2$, another one is the connected sum of two copies of $\mathbf {CP}^2$ with different ...
2
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0answers
107 views

Surgery and Curvature on Foliation

Let $X$ be an oriented closed smooth $4$-manifold. Suppose that $TM$ admits a foliation $\mathcal F$ of dimension two, and admits a positvescalar curvature. Q: If we do the surgery on $X$ to reduce ...
7
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3answers
1k views

About Simon Donaldson's book on four dimensional manifold

Recently I'm reading Donaldson's Geometry of four manifolds. It seems to me that the book requires a lot for background. Additionally, the proof in the book is too sketchy without too much detail. I ...
24
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0answers
1k views

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)...
5
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0answers
438 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 ...
21
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0answers
255 views

Are there exotic $\mathbb{R}^4$'s that are products with $\mathbb{R}$? [duplicate]

This came up in a conversation: Question: Is there an exotic $\mathbb{R}^4$ that smoothly splits off an $\mathbb{R}$ factor? More precisely, suppose that $\mathcal R$ is a smooth 4-manifold which ...
7
votes
1answer
520 views

Surgery along an embedded surface in a 4-manifold

Let $X$ be a 4-manifold and $\Sigma_g\subset X$ be an embedded closed orientable genus = $g$ surface. Suppose $\Sigma_g\subset X$ has a trivial closed normal bundle $N(\Sigma_g) = \Sigma_g\times D^2$. ...
5
votes
2answers
330 views

How can Wall's theorem be generalised to non-simply connected manifolds?

In a sense, this is a follow-up to this question. By work of Freedman and Wall, it is known that if two simply-connected 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathbb{N}$ ...
0
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1answer
172 views

Visualizing the 4th dimension [closed]

In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as shadows of 4-D objects. How does this helps us visualize 4-D objects? I searched that we can atleast see ...
5
votes
1answer
221 views

Fundamental groups of open sets in $R^n$ with $n=3,4$

It is well known that every finitely presented group may be realised as fundamental group of some closed $4$-manifold. What groups can be obtained as fundamental groups of open subsets of $R^4$? I'...
9
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1answer
576 views

Is a 4-dimensional submanifold of a spin manifold always spin?

Let $M^d$ be a $d$-dimensional orientable spin manifold, and $N^4$ is a closed $4$-dimensional orientable submanifold of $M^d$. Is $N^4$ always spin? If $d=5$, is $N^4$ always spin? If $N^4$ is a ...
7
votes
1answer
503 views

Is there a four-manifold whose tangent bundle is an endomorphism bundle?

Is there a smooth four-manifold $M$ such that $TM \cong \operatorname{End}(E)$ for some rank $2$ bundle $E \to M$? If $M$ is parallelisable, then one can take $E$ to be the trivial rank $2$ bundle ...
27
votes
1answer
2k views

Not all manifolds can be triangulated: In which dimensions?

I know that Ciprian Manolescu has settled the triangulation conjecture in the negative: Not all manifolds can be triangulated. I've only read secondary literature on this result, which did not detail ...
12
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2answers
397 views

Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?

It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which ...
3
votes
1answer
238 views

4-dimensional TQFT with/without requiring spin structure

My questions here are focused on $D$-dimensional topological quantum field theories (TQFTs) which are unitary and which have finite dimensional Hilbert space on a closed spatial manifold $M^{d-1}$. ...