The 4-manifolds tag has no wiki summary.

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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 ...

**17**

votes

**1**answer

813 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 ...

**7**

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**1**answer

554 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

595 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 ...

**12**

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**1**answer

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### 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 ...

**2**

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**2**answers

361 views

### Reference for the proof of this statement?

Can anyone give me the reference for this statement?:
Let $M$ be a closed oriented smooth 4-manifold. Any element $a\in H_2(M)$ can be represented by a smoothly embedded, oriented surface.
I found ...

**21**

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**4**answers

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### 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 ...

**7**

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**2**answers

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### 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 ...

**3**

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**4**answers

430 views

### Compactify S^2\times S^2-\Delta

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ ...

**12**

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**1**answer

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### 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 ...

**20**

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**3**answers

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### 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 ...

**9**

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**3**answers

652 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 ...

**61**

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**8**answers

6k 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 ...

**5**

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**3**answers

331 views

### PD3 groups and PD4 complexes

I am interested at the moment in what groups can occur as the fundamental group of a 4-manifold (or more generally, a 4-dimensional CW complex) with prescribed conditions on the intersection form. I ...

**13**

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**2**answers

950 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 ...

**7**

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**2**answers

556 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, ...

**4**

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**4**answers

593 views

### 4-genus of a 2-bridge link

How can we calculate the 4-genus of a link L? The 4-genus is defined to be the minimal genus of orientable surface bounded by L in B^4. Is there any routine method to calculate that?
Especially, any ...

**0**

votes

**1**answer

246 views

### If the 4-genus of a link is zero, is it a slice link?

An n-component slice link is a link that bounds n disjoint discs in B^4. And the 4-genus of a link is defined to be the minimal genus of orientable surfaces bounded by it in B^4.
My question is: if ...

**18**

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**0**answers

668 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 ...

**21**

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**3**answers

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

Real projective spaces ℝPn have ℤ/2 cohomology rings ℤ/2[x]/(xn+1) and total Stiefel-Whitney class (1+x)n+1 which is 1 when n is odd, so it follows that odd dimensional ones are boundaries of compact ...

**26**

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**3**answers

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### 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 ...

**6**

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**2**answers

763 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 ...