All Questions
13 questions
25
votes
1
answer
1k
views
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
- 439
23
votes
13
answers
7k
views
What should be taught in a 1st course on smooth manifolds?
I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic ...
14
votes
2
answers
892
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
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 $\...
- 5,529
5
votes
2
answers
609
views
Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology
Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology.
The evaluation map $$ev\colon N\times\mathcal{C}^\...
- 191
5
votes
1
answer
380
views
Non-density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...
- 5,405
5
votes
0
answers
154
views
Sheaf-like reconstruction of a continuous function
Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\...
- 5,405
3
votes
0
answers
221
views
Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]
Please let me denote the following
(TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
(PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
(PL) ...
- 2,147
2
votes
1
answer
308
views
Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)
every smooth manifold can be triangulated, is it true for orbifold? Is it a known result? If yes, is there any reference?
reply to the comment : G does not need to be any subgroup of Sn , any ...
- 21
2
votes
0
answers
414
views
$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
2
votes
0
answers
74
views
Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
- 6,962
1
vote
0
answers
159
views
How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?
Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...
- 841
0
votes
1
answer
101
views
A question on relation of different triangulations of a triangulable space
Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these ...
- 781