All Questions
11 questions
8
votes
1
answer
224
views
Can increasing the winding number of a 2-cell make a CW complex embeddable?
Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$.
For a natural number $n\ge 2$ consider the operation of ...
- 976
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 ...
- 161
16
votes
3
answers
2k
views
When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
- 13.6k
0
votes
1
answer
135
views
Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
- 508
4
votes
2
answers
374
views
Knot theory in handlebodies of arbitrary genus
It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...
- 28.2k
9
votes
3
answers
1k
views
Contractibility of space of embeddings of a disc
I'm pretty sure that both of the following spaces are contractible. However, I can't seem to find a proof or a reference. Can anyone provide one? Let $D^2$ be the unit disc in $\mathbb{R}^2$.
The ...
- 35.5k
20
votes
3
answers
2k
views
Homotopy groups of spaces of embeddings
Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold $M$ into a manifold $N$ equipped with smooth compact-open topology.
Question 1. Are there conditions ensuring that ...
- 2,974
25
votes
1
answer
2k
views
On a curious map from the complex projective plane into $S^5$
I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
- 5,215
4
votes
1
answer
220
views
Embedding spaces and surface knots in high dimensional manifolds
This is a variation of Craig's Knot complement diffeomorphism groups and embedding spaces for a different type of very simple manifold (surfaces which have a 1-relator fundamental group instead of ...
- 18.8k
9
votes
1
answer
384
views
embedding of quaternionic projective spaces
Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...
- 666
2
votes
1
answer
1k
views
embeddings of product of spheres in Euclidean spaces [closed]
I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m_1}\times\cdots\times S^{...
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