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.
30
questions
39
votes
1
answer
6k
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 ...
- 149k
27
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 ...
- 10.2k
37
votes
3
answers
993
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. ...
- 735
17
votes
1
answer
699
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 ...
- 5,545
14
votes
2
answers
3k
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 ...
- 4,761
10
votes
1
answer
583
views
Index of Dirac operator and Chern character of symmetric product twisting bundle
I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...
94
votes
4
answers
12k
views
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 what'...
- 28.7k
45
votes
1
answer
2k
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 $...
- 9,748
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 ...
- 3,831
24
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 3-...
24
votes
0
answers
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)...
- 1,619
17
votes
1
answer
2k
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 ...
- 10.2k
17
votes
1
answer
931
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 $H^2(M;\mathbb{Z})/\...
- 193
16
votes
1
answer
3k
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 ...
- 28.8k
15
votes
1
answer
431
views
Status of a conjecture of C.T.C. Wall?
In Wall's paper Unknotting tori in codimension one and spheres in codimension two, he states the following conjecture:
Any $h$-cobordism of $S^3 \times S^1$ to itself is diffeomorphic to $S^3 \times ...
- 637
13
votes
2
answers
689
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 ...
- 5,545
12
votes
1
answer
740
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 ...
- 12.8k
12
votes
1
answer
501
views
Unknotted $S^{n-2}$ in $S^n$
I wonder is it still an open question that a smooth sphere $\Sigma^{2}\subset S^4$ is unknotted in $S^4$ iff its complement is homotopy equivalent to $S^1$? If it is an open question, how is it ...
- 1,705
10
votes
2
answers
2k
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 $N$ ...
- 921
10
votes
1
answer
744
views
What are Kirby diagrams of candidate exotic 4-manifolds?
It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle ...
- 5,545
8
votes
1
answer
2k
views
Relation of SW and Donaldson Invariant
My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space ...
- 693
8
votes
1
answer
598
views
Are all 4-manifolds $Pin^{\tilde{c}}$?
It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$.
A usual generalization of this ...
- 575
8
votes
1
answer
650
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 $N(\Sigma_g) = \Sigma_g\times D^2$. ...
6
votes
1
answer
516
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 ...
- 5,319
6
votes
1
answer
482
views
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 $S^1 \times D^3$ ...
- 773
6
votes
1
answer
304
views
Computation of $\pi_1$ for a Mazur manifold and its boundary
If we attach a $4$-dimensional $1$-handle $D^1 \times D^3$ to a $4$-dimensional $0$-handle $B^4$, we obtain $S^1 \times D^3$. The null homologous knot in $S^1 \times S^2$ indicated in the picture ...
- 606
5
votes
2
answers
622
views
Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?
It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...
- 5,545
5
votes
1
answer
284
views
Stable torus that is not a torus [duplicate]
Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?
- 43.7k
4
votes
1
answer
246
views
0-homologous surface bounds
Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented 3-...
- 2,283
3
votes
1
answer
402
views
Simply-connected 4-manifolds can be blown up and down to complex projective planes. How about non-simply-connected ones?
There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$:
$$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \...
- 5,545