The 4-manifolds tag has no usage guidance, but it has a tag wiki.

**71**

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### How can there be topological 4-manifolds with no differentiable structure?

This is a very naive question, and I'm hoping that it will be matched by a correspondingly elementary answer. It is well known that not every topological 4-manifold admits a smooth structure. So ...

**63**

votes

**9**answers

7k views

### Theoretical physics: Why not just R^4?

You and I are having a conversation:
"Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."
"Something like that"
"...And ...

**30**

votes

**0**answers

1k views

### 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.
...

**27**

votes

**3**answers

2k views

### What manifolds are bounded by RP^odd?

Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...

**27**

votes

**3**answers

1k views

### Embeddings of $S^2$ in $\mathbb{CP}^2$

Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect ...

**24**

votes

**1**answer

1k views

### Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$
Background
Definition (Exotic $\mathbb{R}^4$):
An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...

**22**

votes

**4**answers

2k views

### Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...

**22**

votes

**3**answers

2k views

### How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]

With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about ...

**22**

votes

**1**answer

1k 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 ...

**19**

votes

**0**answers

663 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 ...

**18**

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**0**answers

504 views

### 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 ...

**18**

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**0**answers

721 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 ...

**17**

votes

**1**answer

890 views

### What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$?
Background
Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...

**15**

votes

**1**answer

585 views

### Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on ...

**14**

votes

**3**answers

1k views

### open problems in Seiberg-Witten Theory on 4-Manifolds

What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...

**14**

votes

**4**answers

1k views

### How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...

**14**

votes

**2**answers

1k views

### topological “milnor's conjecture” on torus knots.

Here's a question that has come up in a couple of talks that I have given recently.
The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly ...

**14**

votes

**1**answer

695 views

### Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?

Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper.
...

**14**

votes

**1**answer

306 views

### Shortest Casson tower containing a slice disk for the attaching curve

A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk ...

**13**

votes

**2**answers

2k views

### What does Yang-Mills and mass gap problem has to do with mathematics?

I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and ...

**13**

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**1**answer

1k views

### A question on classification of almost complex structures on $4$-manifolds

I have a (basic?) question in topology.
Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by ...

**13**

votes

**1**answer

401 views

### Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...

**13**

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**1**answer

1k views

### Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)

A celebrated theorem of Rohlin states the following
An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero.
Simple homological arguments ...

**12**

votes

**1**answer

529 views

### Handlebody decomposition of an open 4-manifold

Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth ...

**12**

votes

**2**answers

657 views

### Explicit embeddings of Cappell-Shaneson knots

In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare ...

**12**

votes

**0**answers

368 views

### Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$.
Here we denote by $s(K)$ Rasmussen's s-invariant for $K$,
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a ...

**11**

votes

**1**answer

1k views

### Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?

**11**

votes

**2**answers

573 views

### Existing proofs of Rokhlin's theorem for PL manifolds

I'm looking for a comprehensive reference to existing proofs of Rokhlin's theorem that a 4-dimensional closed spin PL manifold has signature divisible by 16.
I'm specifically interested in direct ...

**11**

votes

**1**answer

732 views

### Thurston geometries in dimension 4

In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3.
Question: How many different geometries (in the sense of Thurston) do we have in ...

**11**

votes

**1**answer

578 views

### Smooth four-manifolds with contractible universal cover

Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible?
It semms to me that the answer is negative when $X$ is simply ...

**11**

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**0**answers

226 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 ...

**10**

votes

**2**answers

455 views

### Embedding the product of three circles in the 4-sphere.

Let $M$ be a smooth submanifold of the 4-sphere $S^4$. I'm going to demand that $M$ be diffeomorphic to $S^1 \times S^1 \times S^1$. By Jordan-Brouwer separation, $M$ separates the 4-sphere into ...

**10**

votes

**1**answer

307 views

### Piecewise linear (PL) structures on $\mathbf R^4$

One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...

**10**

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**1**answer

271 views

### Visualising locally flat embeddings of surfaces in R^4

As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally ...

**10**

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**0**answers

157 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 $S^4$. ...

**9**

votes

**2**answers

308 views

### Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$

In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...

**9**

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**2**answers

1k views

### Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read
"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is ...

**9**

votes

**3**answers

730 views

### Alexander polynomial or Reidemeister torsion for knotted surfaces?

An important invariant of a knot in $S^3$ is its Alexander polynomial, related also to Reidemeister torsion. Is there something like that for knotted surfaces in $S^4$? If not, what are the ...

**9**

votes

**1**answer

415 views

### Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively

I'm interested in a complexity question related to problems like the slice-ribbon problem.
To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...

**8**

votes

**2**answers

905 views

### When is the connected sum of manifolds orientation-independent?

Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed?
If $N$ ...

**8**

votes

**1**answer

171 views

### Essential Klein bottle in simply connected symplectic 4 manifolds

Consider the following question:
Let $X$ be a simply connected, symplectic 4-manifold. Does there exists a smoothly embedded Klein bottle $K\subset X$ such that the following conditions are both ...

**7**

votes

**1**answer

740 views

### A remark by Gromov on 4-manifolds

Gromov remarks in a a survey on manifolds (p.12) that "it is hard to imagine that there are infinitely many non-diffeomorphic, but mutually homeomorphic, quotients of the hyperbolic 4-space by ...

**7**

votes

**1**answer

295 views

### Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin.
In 4 dimensions, $CP^2$ is not spin. I would like to ask if
all 4-dimensional compact orientable mapping tori are spin?
See ...

**7**

votes

**1**answer

561 views

### Are there symplectic 4-folds with $b_+>1$, $b_-=0$?

This is the question. Is it known that a symplectic $4$-fold with $b_2>1$ should have a homology class $C$ with $C^2<0$?

**7**

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**1**answer

695 views

### Gromov-Witten invariants counting curves passing through two points

Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form
$GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in ...

**7**

votes

**2**answers

610 views

### slice=ribbon generalization to higher genus + potential counterexamples to slice=ribbon.

I have two questions about the slice=ribbon conjecture.
(1) If a knot $K \hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 \times [0, ...

**6**

votes

**2**answers

817 views

### A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve?

There's an old problem in 4-manifold theory that, as far as I know, doesn't have a name associated with it and really deserves a name.
Let $M$ be a smooth 4-manifold with boundary. Let $S$ be a ...

**6**

votes

**1**answer

389 views

### Almost parallelizable 4-manifolds

On the first page of Milnor-Kervaire's paper "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", they assert without proof or reference that if $M$ is a compact connected oriented ...

**6**

votes

**2**answers

276 views

### Knotted projective planes and fake complex projective space

Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is ...

**6**

votes

**1**answer

263 views

### Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
...