All Questions
35 questions
3
votes
0
answers
119
views
Signature vs commensurability
If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...
- 41
25
votes
1
answer
582
views
Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 587
5
votes
1
answer
380
views
Proving the Cork Theorem
I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
13
votes
1
answer
518
views
Low dimensional homotopy groups of $\operatorname{Top}(4)$
$\DeclareMathOperator\Top{Top}$I would like to compute $\pi_3\Top(4)$ and $\pi_4\Top(4)$. It is known that $\Top(4)/O(4) \rightarrow \Top/O$ is 5-connected and
$$
\pi_k(\Top/O) =
\begin{cases}
...
3
votes
0
answers
637
views
What are some of the big open problems in $4$-manifold theory?
I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...
- 139
4
votes
1
answer
421
views
4-manifold $M$ with intersection form of Leech lattice
Do we know any 4-manifold $M$ with intersection form of rank-24 Leech lattice?
Is $M$ smooth? Differentiable to which $n$-order?
Can $M$ be obtained by any surgery on 3 copies of E8 4-manifolds with ...
- 447
6
votes
1
answer
304
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?
- 45k
2
votes
0
answers
197
views
$4$-manifolds with boundary homotopic to $K(G,1)$
I am not very conversant with the $3$-manifold or $4$-manifold theory. The literature I found so far related to $3$-manifold tells me that $S^1\times S^2$ is "very special". It is prime but ...
- 1,177
6
votes
1
answer
324
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
7
votes
2
answers
617
views
Status of the Hopf-Thurston sign conjecture in dimension 4
A famous conjecture in topology asserts:
The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.
This was conjectured by Hopf for manifolds with non-...
- 11.9k
11
votes
0
answers
335
views
Isotopy on embedded 3-manifolds in 4-manifolds
Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
- 3,751
1
vote
0
answers
160
views
Contractible four-manifold which admits a decomposition
Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is ...
- 891
6
votes
1
answer
254
views
Contractible 4-manifold with an exhaustion
Let $M^4$ be a noncompact, contractible, 4-dimensional, topological manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) ...
- 1,441
6
votes
0
answers
162
views
Can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable, PL or DIFF manifold, if $X_4$ is a non-triangulable manifold? [duplicate]
Question: If $X_4$ is a non-triangulable topological (TOP) manifold,
can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable manifold?
can $X_4 \times S^1$, $X_4 \...
- 10.5k
10
votes
2
answers
659
views
Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?
Recall that two 4-manifolds $M$ and $N$ are stably diffeomorphic if there exist $m,n$ such that
$$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$
That is, they become diffeomorphic after ...
- 27.5k
14
votes
2
answers
1k
views
Very particular kind of 4-manifolds. Classification
Let $M$ be a smooth orientable compact connected (with boundary) manifold of dimension $4$. In addition $M$ is assumed to be aspherical and acyclic.
Question: is there a "classification" of ...
- 223
17
votes
2
answers
1k
views
Homotopy groups of Diff(X) and Homeo(X)
For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general ...
- 19.4k
10
votes
2
answers
709
views
4-dimensional cohomology $\mathbb{CP}^2$'s
Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?
- 6,995
8
votes
0
answers
489
views
Self diffeomorphism of $S^2\times S^2$
The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2
Is it true that every self ...
- 3,751
9
votes
1
answer
472
views
Topological mapping class groups of 4-manifolds
It is a classical result of Quinn that for a simply-connected
closed $4$-manifold $X$ the isometries of its intersection form
are in one-to-one correspondence with
$\pi_0 \text{Homeo}(X)$. (Isotopy ...
- 135
3
votes
0
answers
405
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-...
- 10.5k
25
votes
1
answer
2k
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 ...
- 3,751
8
votes
1
answer
686
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 $N(\Sigma_g) = \Sigma_g\times D^2$. ...
41
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 ...
- 151k
5
votes
1
answer
232
views
$\pi_1$ 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 ...
- 773
14
votes
2
answers
1k
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.
...
17
votes
1
answer
981
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
5
votes
1
answer
474
views
Casson invariant and signature
In W. Neumann, J. Wahl, "Casson invariant of links of singularities",
Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...
- 743
4
votes
1
answer
492
views
Boundaries of smooth manifolds
If one has a smooth simply connected manifold $M^n$ which we know to bound an $n+1$ dimensional manifold $N$, what can be said about a handle decomposition for one in terms of a handle decomposition ...
- 41
16
votes
3
answers
3k
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 ...
8
votes
1
answer
734
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 ...
- 155
10
votes
1
answer
489
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 ...
- 14.3k
5
votes
2
answers
1k
views
Intersection forms of 4-manifolds with boundary
Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
- 53
11
votes
2
answers
844
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 ...
- 7,842