All Questions
Tagged with gt.geometric-topology gn.general-topology
86 questions with no upvoted or accepted answers
2
votes
0
answers
115
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Direct limits of compact surfaces with uniformly bounded topology
Suppose we have a directed system of inclusions of compact surfaces with boundary
$$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \cdots $$
such that all of the surfaces $\{S_k\}$ have uniformly bounded ...
- 1,601
2
votes
0
answers
72
views
Topological Shape Operator More Sensitive than Inverse Limits
This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
- 439
2
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0
answers
82
views
Enveloping a Jordan curve with a trace of another one
This question is inspired by this one, or rather the way I understood it.
Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
- 5,529
2
votes
0
answers
98
views
Union of Two Faces, using the Jordan Curve Theorem
Consider four disjoint points in the plane, $v_{1}$, $v_{2}$, $v_{3}$ and $v_{4}$.
The cycle, $C:=v_1v_2v_{3}v_{4}v_{1}$, is the union of the (Jordan)
arcs, $A_{12}$, $A_{23}$, $A_{34}$, and $A_{41}$, ...
- 21
2
votes
0
answers
109
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Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms
We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).
Given an embedded torus $T$ with trivial normal ...
- 10.5k
2
votes
0
answers
224
views
cross-sections of a sphere bundle
Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...
- 2,223
2
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0
answers
87
views
Local section of Lie Groupoids
Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
- 21
2
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0
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212
views
Can a compact metrizable space be determined by its Hausdorff measures?
Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\...
2
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0
answers
192
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Are open convex PL subsets of R^n PL homeomorphic to R^n?
This is a basic issue of PL topology that I assume must be true, but I can't find a written reference: is a convex open PL subset of $\mathbb R^n$ PL homeomorphic to $\mathbb R^n$? I've scanned ...
- 5,534
1
vote
0
answers
48
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Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
user530766
1
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0
answers
61
views
Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"
Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
- 159
1
vote
0
answers
145
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Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
- 447
1
vote
0
answers
141
views
Can a closed null-homotopic curve be filled in by a disc?
Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
- 13.6k
1
vote
0
answers
97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
1
vote
0
answers
143
views
End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
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$ ...
- 11
1
vote
0
answers
298
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
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 ...
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 $...
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$-...
- 19
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
1
vote
0
answers
126
views
Determine all possible magnetic monopole of gauge theories
In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...
- 10.5k
1
vote
0
answers
132
views
When is a nested sequence of closed sets a colimit?
Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...
- 1,174
1
vote
0
answers
105
views
The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11
1
vote
0
answers
152
views
Stone cech compactification of a zero dimensional topological space
Let $X $ be a zero dimensional topological space, that is, a topological space with a basis of clopen sets. Is there any characterization for the ston cech compactification for such a space?
- 11
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
1
vote
0
answers
145
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Reference needed: Does pseudo laminated compact subsets of the plane separate the plane?
Hi,
doing my research I found the following problem and I´ll be glad if someone could give a reference.
We say that a compact connected subset $K$ of the plane is psuedo laminated if the following ...
- 19
1
vote
0
answers
365
views
Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
0
votes
0
answers
128
views
The smallest dihedral angle of convex polyhedrons
Given a set of points $\{x_{k}\}_{k=0}^{m} \subset \mathbb{R}^n$, is it always possible to find a constant $c=c(m,n)>0$, depending only on the dimension $n$ and the number $m$, such that, after ...
- 705
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. ...
0
votes
0
answers
74
views
Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
- 9
0
votes
0
answers
78
views
Pareto-optimal front $F$ in $m$-dimensional space can not have more than $\mathbf{H}_{m-2}(F)$ homology groups
I need to prove that a Pareto-optimal front $F$ in $m$-dimensional space (i.e. $m > 1$) can not have more than $\mathbf{H}_{m-2}(F)$ homology groups.
What it simply means that in a 2-dimensional ...
- 101
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 ...
0
votes
0
answers
850
views
Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
- 1
0
votes
0
answers
365
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Finding paths in a path connected space
I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...
- 627