All Questions
25 questions
9
votes
0
answers
258
views
Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...
- 2,806
7
votes
2
answers
534
views
Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
- 605
8
votes
1
answer
380
views
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
- 725
0
votes
0
answers
177
views
Homeomorphism groups on manifolds and topological properties
Let $M$ be a compact $n$-dimensional manifold let $H(M)$ denote the homeomorphism group of $M$.
If $n=2$ then $H(M)$ enjoys nice properties such as being an ANR, is locally contractible, separable. ...
3
votes
1
answer
325
views
A detail in Brown's proof of the generalized Schoenflies theorem
Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote
$$S^{n-1}_t=h(S^{n-1}\times \{t\}).$$
The generalized Schoenflies theorem states the closure of each connected ...
- 1,225
8
votes
1
answer
233
views
If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?
Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.
Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \...
- 13.6k
4
votes
1
answer
236
views
What is the Freudenthal compactification of a wildly punctured n-sphere?
Let $C$ be a compact and totally-disconnected subspace of the $n$-sphere $\mathbb{S}^n$, where $n\geq 2$.
Question: Must the Freudenthal compactification of $\mathbb{S}^n \setminus C$ be homeomorphic ...
- 1,926
14
votes
2
answers
889
views
Must a space that is locally injective image of $\mathbb{R}^n$ be a manifold?
Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is ...
- 623
7
votes
1
answer
354
views
Decomposition of manifolds with toroidal boundary
Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following this article, it is stated that $\mathcal{M}$ can be written as
...
- 1,429
14
votes
1
answer
937
views
Classification of 3-dimensional manifolds with boundary
It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as
$$\mathcal{M}=P_{1}\#\dots\# P_{n}$$
where $P_{i}$ are prime manifolds, i.e. ...
- 1,429
6
votes
1
answer
506
views
Map which is null-homotopic on compacts
This is the missing ingredient towards answering my previous question.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
- 5,529
9
votes
2
answers
755
views
Is limit of null-homotopic maps null-homotopic?
The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\...
- 5,529
5
votes
1
answer
380
views
Non-density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...
- 5,405
3
votes
0
answers
221
views
Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]
Please let me denote the following
(TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
(PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
(PL) ...
- 2,147
8
votes
1
answer
1k
views
Cobordism Theory of Topological Manifolds
Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.
Cobordism Theory for DIFF/Differentiable/smooth manifolds
However, there are ...
- 10.5k
14
votes
1
answer
578
views
Obstruction of spin-c structure and the generalized Wu manifods
Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the
$$
H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
- 2,147
8
votes
2
answers
2k
views
Any 3-manifold can be realized as the boundary of a 4-manifold
We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
- 10.5k
17
votes
1
answer
525
views
Lowest Dimension for Counterexample in Topological Manifold Factorization
Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...
- 439
7
votes
1
answer
731
views
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 ...
- 6,881
5
votes
1
answer
258
views
Generating the topology of a manifold
Let $X$ be a topological manifold of dimension $d$, and let $F$ be a collection of continuous maps from $X$ into $\mathbf{R}^d$ such that:
$F$ separates points of $X$, i.e. for any two distinct ...
- 5,529
16
votes
4
answers
2k
views
Self-covering spaces
Let $M$ be a connected Hausdorff second countable topological space. I will call $M$ self-covering if it is its own $n$-fold cover for some $n>1$. For instance, the circle is its own double cover ...
- 1,959
40
votes
1
answer
2k
views
Are there only countably many compact topological manifolds?
Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...
- 3,017
19
votes
4
answers
4k
views
When is a finite cw-complex a compact topological manifold?
I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
- 732
18
votes
5
answers
2k
views
Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
I'm curious about the following:
Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...
- 761
7
votes
1
answer
789
views
Counting submanifolds of the plane
After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane.
My ...
- 28.2k