# 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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### 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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**1**answer

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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 ...

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### 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 \\
...

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159 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 $S^4$?

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### 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 ...

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**1**answer

157 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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118 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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**1**answer

197 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). ...

**33**

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**1**answer

626 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 $...

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### 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 ...

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**1**answer

93 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 ...

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907 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.

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**1**answer

568 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 ...

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**2**answers

489 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; \...

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### 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.

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**1**answer

261 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 ...

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**1**answer

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### 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 ...

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### 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. ...

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votes

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### 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. ...

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votes

**2**answers

342 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,\...

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**1**answer

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### 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 ...

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269 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. ...

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315 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 ...

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### 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 ...

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votes

**3**answers

969 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 ...

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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)...

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### 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 ...

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### 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}^...

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### 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 ...

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**1**answer

500 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 $N(\Sigma_g) = \Sigma_g\times D^2$. ...

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### 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 $k \in \mathbb{N}$ ...

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**1**answer

159 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 ...

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**1**answer

203 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'...

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538 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 ...

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votes

**1**answer

481 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 ...

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**1**answer

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### 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 ...

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votes

**2**answers

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### 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 ...

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**1**answer

219 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 $M^{d-1}$. ...

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### transversality for reducible holomorphic curve

Let $(X, \omega)$ be a strong symplectic cobordism between nondegenerate contact manifolds $(Y_+, \lambda_+)$ and $(Y_-, \lambda_-)$. Let $\overline{X}$ be usual symplectization completion of $X$ ...

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### Evolution of $W_+$ and $W_-$ under the Ricci flow

In dimension $4$ the Weyl operator $W$ splits in two parts
$$W_+:\Lambda^{2}_{+} \to \Lambda^{2}_{+}$$
and
$$W_-:\Lambda^{2}_{-} \to \Lambda^{2}_{-}.$$
(a) Has there been a study of the evolution ...

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**1**answer

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### $\pi_1$ of 4-manifolds that “look like” disk bundles

Let $X$ be a smooth compact oriented 4-manifold with $\partial X=L(p,1)$, $H_2(X;\Bbb Z)=\Bbb Z$, $H_3(X; \Bbb Z)=0$ and the induced map $\pi_1(L(p,1)) \to X$ surjective. What are the possibilities ...

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### A link of four 2-tori $T^2$ in $S^2 \times S^2$

Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-...

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### Simply connected 4-manifolds with boundary

I think I've encountered a question about 4-manifolds which maybe easy but I'm not familiar with. Can anyone give me an example of a simply connected 4-manifold $M$ (with boundary, of course) with $...

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### 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-...

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**1**answer

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### Distinguishing homology $S^1 \times S^2$'s which bound homotopy $S^1$'s

Due to Mazur, Akbulut and Kirby and many others, there are many examples of integer homology 3-spheres which bound contractible 4-manifolds given by attaching a single 2-handle to $S^1 \times D^3$ ...

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**1**answer

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### A proof of Theorem 9.2.12. in the Gompf-Stipsicz

I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a ...

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**2**answers

296 views

### CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...

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**1**answer

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### How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...

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**1**answer

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### Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...