All Questions
Tagged with gn.general-topology gt.geometric-topology
329 questions
3
votes
0
answers
282
views
Commutator length of the fundamental group of some grope
A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra
$L_0 \to L_1 \to L_2 \to \cdots$
obtained as follows. Take $L_0$ as some $S_g$, an ...
- 2,083
4
votes
1
answer
522
views
Can every manifold be represented as a quotient
My question is "inspired" by the uniformization theorem for Riemmannian surfaces and this post.
Suppose that $X$ is connected (finite-dimensional) topological manifold without boundary. ...
- 6,367
1
vote
1
answer
102
views
Approximate Jordan-Brouwer theorem (corrected)
My first attempt to ask this question sort of failed (I'll explain below).
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{...
- 56.9k
2
votes
1
answer
131
views
Approximate Jordan-Brouwer theorem
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is ...
- 28k
6
votes
0
answers
196
views
Logarithm on formal power series continuous?
Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...
- 463
5
votes
1
answer
372
views
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?
Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...
- 10.5k
1
vote
1
answer
141
views
Action of the permutation group on the set of topologies on a continuum
Let $X$ be a set of continuum cardinality. The group of permutations of $X$ acts on the set of topologies on $X$.
What can be said about the fixed points of this action? Are manifolds fixed points?
...
- 63.9k
9
votes
1
answer
321
views
Isotopies, Fiber Bundles and Selection Theorems
The following problem is a culmination of a few questions I've asked the last two months, and it's still giving me some issues. I think I know the right way to solve it, but I'm having trouble with ...
- 439
1
vote
0
answers
77
views
Given a homeomorphism on $\mathbb{R}^3$, can its effects on a compact subset be realized by a homeomorphism that's non-identity only on a compact set?
Let $f_1 \colon \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 \colon \mathbb{R}^3 \to \mathbb{R}^3$ ...
- 66.3k
1
vote
0
answers
297
views
Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 673
0
votes
1
answer
210
views
Questions on the proof Lemma 4.5 GTM 175, Lickorish
I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given.
For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\...
- 1,617
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,\...
- 63.9k
6
votes
2
answers
324
views
Nonvanishing section of infinite-dimensional tautological bundle
Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have ...
- 7,583
0
votes
0
answers
140
views
Can the infinite jungle gym surface be expressed by an exhaustion of compact surfaces with one boundary component?
Is it possible to write the infinite jungle gym surface as the increasing union of compact surfaces whose boundaries consist of only one simple closed curve? I believe that this is true. This surface ...
7
votes
1
answer
183
views
Stability Question for Isotopies Between Compact Sets
Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.
Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
- 12.3k
0
votes
1
answer
277
views
Are knot invariants topological invariants? [closed]
I am a bit confused about terminology considering topology and knot theory.
A topological invariant is considered to be a topological property that does not change under a homeomorphism of the space.
...
- 3,065
1
vote
0
answers
80
views
A characterization for a space that is similar to locally connected spaces
Let $X$ be a $T_0$ topological space with the property that there exists a basis $\{O_i\}_{i\in I}$ for $X$ such that for each $J\subseteq I$ the subspace $\bigcap_{i\in J}O_i$ has only finitely many ...
- 12.9k
1
vote
1
answer
248
views
The Schoenflies Theorem on two dimensional surfaces
Let $S$ be a surface and $U$ an open connected subset of $S$. If the frontier of $U$ in $S$ is a two sided circle $C$, then the closure of $U$ in $S$ is a surface whose boundary is $C$. I would like ...
- 1,617
6
votes
1
answer
300
views
Proof of Denjoy-Riesz Theorem and Moore's Generalization?
The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
- 439
21
votes
7
answers
1k
views
Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
- 188k
1
vote
0
answers
225
views
Is it possible to prove that surfaces with compact boundary are homeomorphic by glueing disks to the boundary components?
Let $S_1$ and $S_2$ be two surfaces with compact boundary and the same number of boundary components. Let $M_1$ and $M_2$ be the surfaces obtained by glueing closed disks to the boundary of $S_1$ and $...
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 ...
- 45k
9
votes
2
answers
245
views
Toroidal Heegaard splittings
Suppose I have a Heegaard splitting of a closed oriented irreducible 3-manifold $M$, defined by the Heegaard diagram $(\Sigma_{g},\{\alpha_{1},\dots,\alpha_{g}\},\{\beta_{1},\dots,\beta_{g}\})$. Are ...
- 459
5
votes
1
answer
409
views
Statements related to Thurston's work on the surface
If we have simple closed curves $\alpha$ and $\beta$ on a surface $\Sigma_g$, the intersection number $i(\alpha ,\beta)$ is defined to be the minimal cardinality of $\alpha_1\cap\beta_1$ as $\alpha_1$ ...
- 489
1
vote
1
answer
423
views
Extension of homeomorphisms
Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical ...
- 8,167
2
votes
1
answer
94
views
Density of functions into the circle glueing
Let $\{U_i\}_{i=1}^2$ be an open cover of $S^1$, with $U_i\cong \mathbb{R}$ (for example, $U_1$ is the lower arc of the circle and $U_2$ is the upper part). Let $\iota_i:U_i\hookrightarrow S^1$ be ...
- 16.4k
4
votes
1
answer
258
views
Density of compactly-supported homeomorphisms
**Disclaimer:**I posted the following question on MSE, but since there were no answers. I'm migrating it here.
Let $Homeo_0(\mathbb{R}^n)$ ($Homeo_c(\mathbb{R}^n)$) be the space of all (compactly-...
- 18.2k
2
votes
0
answers
323
views
Continuous injective functions with dense image
Let $X$ be the set of continuous, injective functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ with dense image; and equip $X$ with the (relative) compact-open topology. What is known about this space?
...
- 5,405
10
votes
2
answers
451
views
Group of surface homeomorphisms is locally path-connected
I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance ...
- 1,517
15
votes
3
answers
1k
views
What do absolute neighborhood retracts look like?
In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...
- 1,869
77
votes
4
answers
15k
views
What are good mathematical models for spider webs?
Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking “if you understand the spider web, you understand the space around it”. ...
- 111
9
votes
0
answers
333
views
Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
- 117
9
votes
0
answers
200
views
Homotopical characterization of manifolds
Let $X$ be a compact metrizable topological space of covering dimension $4$.
Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...
- 117
35
votes
4
answers
4k
views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
- 1,306
1
vote
0
answers
152
views
Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
- 11
1
vote
0
answers
154
views
Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
- 2,754
1
vote
1
answer
208
views
Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
CommunityBot
- 1
4
votes
0
answers
129
views
'Monodromy' for relative homology group
Let $A$ and $X$ be topological manifolds. Denote by $\mathbb {Emb}(A,X)$ the space of all topological embeddings $A\to X$.
A loop $f_s:A\to X$ ($s\in[0,1]$) in $\mathbb{Emb}(A,X)$ should give rise to ...
- 2,789
3
votes
0
answers
73
views
A holomorphic shrinking of a domain into a compact subset
This question is related to these two.
Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$...
- 5,529
2
votes
0
answers
77
views
Dense embeddings into Euclidean space
The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
- 63.9k
17
votes
3
answers
954
views
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$?
Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be smoothly embedded in $\mathbb R^{2n-1}$?
I know these cannot exist for $n=1$, i.e. $S^...
- 3,751
4
votes
0
answers
228
views
Enlarging a compact set in order to improve its shape
In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $K\subset X$ is compact, then there is a Peano continuum $L\subset X$ such that $K\...
- 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 $\...
- 12.9k
18
votes
1
answer
1k
views
Sperner's Lemma implies Tucker's Lemma - simple combinatorial proof
Sperner’s Lemma is often called the "combinatorial analog" of Brouwer’s Fixed Point Theorem, and similarly Tucker’s Lemma is often called the combinatorial analog of Borsuk–Ulam’s Theorem.
We can ...
- 6,917
5
votes
3
answers
730
views
Is it possible to connect every compact set?
Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set.
Is there a always a compact connected $L\subset X$ such that $K\subset L$?
This ...
- 5,529
7
votes
1
answer
229
views
Retracting off a compact set
Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected.
Can we always find an open $V$ such that $K\subset V\subset\...
- 41.9k
9
votes
4
answers
3k
views
Associativity of topological join and join of spheres
This must be an elementary question, as I couldn't find any proofs on the Internet, but I still can't do it. And yes, Hatcher says that the join is not actually associative for general topological ...
- 3,407
1
vote
0
answers
53
views
Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,529
2
votes
0
answers
208
views
Retracting to a bigger compact
Consider the topological spaces $X$ with the following property:
For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$.
Let ...
- 128k
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). ...
- 959