All Questions
Tagged with gn.general-topology dg.differential-geometry
127 questions
2
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0
answers
58
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Dimension changes from global to local immersion
From Hatcher Corollary A.10. the (global) immersion for an $n-$dimensional CW complex is possible in some $\mathbb{R}^N$. I have started with $M(G,n)$ (Moore space of type $(G,n)$, $G$ is cyclic ...
- 1,177
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
2
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0
answers
263
views
Are these two definitions of smooth $k$-manifold as a Euclidean subset equivalent?
I am struggling to reconcile the two definitions of smooth k-manifold in $R^n$ from M.Spivaks Calculus on Manifolds (pg 109) and J.W Minor's Topology from differential point of view (pg 01).
Milnor's ...
- 21
2
votes
1
answer
152
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How to define "interior" for the unit arc? [closed]
Let the unit arc be,
$$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$
There is something I found curious about the unit arc which is that,
It has an empty interior viewed as a ...
- 179
2
votes
0
answers
35
views
If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
2
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0
answers
171
views
Existence of compatible almost complex structure of symplectic fibration with nice property
Let $\pi: E \to B$ be a fibration over a closed surface $B$ with fier $g(F) \ge 2$. Suppose that both of $B$ and $F$ are closed surfaces and $g(B) \ge2$ and $g(F) \ge 2$. Fix a Kahler structure $(...
- 185
2
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0
answers
305
views
Are homotopy equivalent manifolds with homeomorpic boundaries themselves homeomorphic?
Let $f:M \to M′$ be a homotopy equivalence of topological manifolds with boundary such that $dim(M)=dim(M′)$ and $f:\partial M \to \partial M′$ is a homeomorphism. Does this imply the existence of a ...
- 79
2
votes
0
answers
96
views
Branch point and alexandrov embeddedness
This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow:
Let us assume that $\Sigma_n$ is a sequence of topological ...
- 914
2
votes
0
answers
203
views
Faithful actions of finite groups on topological spaces
Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...
- 15.5k
1
vote
1
answer
284
views
Poincare duality-differential geometry
Let $ M $ be a smooth and compact manifold with boundary $\partial M = X \times F $ on which the structure of a smooth locally trivial bundle $$ \pi: \partial M \longrightarrow X $$
where the $ X $ ...
- 27
1
vote
1
answer
649
views
Local diffeomorphisms, covering maps and smooth path lifting
Let $f: M\to N$ be a surjective local diffeomorphism of noncompact smooth manifolds.
Suppose that every smooth path is liftable, that is, for any smooth path $\gamma: [0,1]\to N$ and any point $p\in f^...
- 2,934
1
vote
1
answer
195
views
Can one explore a surface along ‘piecewise planar’ curves?
Suppose $d\in \{3,4,\dotsc\}$ and $A\subseteq \mathbb{R}^d$ is non-empty, open and connected with its complement $A^c$ connected too and $\text{int}(A^c)\neq \emptyset$. Its boundary $S:=\partial A$ ...
- 1,461
1
vote
1
answer
291
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Isometry and gluing between smooth manifolds - some references
I have a doubt that assails me.
The technique of gluing along edges between manifolds is generally considered in the topological context.
I don't know if there are other gluing techniques.
I was ...
- 272
1
vote
1
answer
248
views
Tightening a loop
Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
- 109
1
vote
1
answer
370
views
Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space
This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...
- 191
1
vote
1
answer
390
views
Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
- 13
1
vote
0
answers
192
views
Simple left earthquakes are dense
i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows:
Definition. If $\lambda$ is a geodesic ...
- 11
1
vote
0
answers
61
views
Minimal radius of a ball admitting a trivialization of a vector bundle
Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...
1
vote
0
answers
92
views
Topological space modeled by special topological structures
Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
- 6,023
1
vote
0
answers
112
views
Question regarding the image of a polynomial map containing a small box
I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously.
Let $\delta, \varepsilon > 0$.
Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
- 3,625
0
votes
1
answer
525
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
- 612
0
votes
0
answers
303
views
Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
- 247
0
votes
0
answers
70
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 247
0
votes
1
answer
284
views
Creating topological spaces with portals [closed]
I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
- 19
0
votes
0
answers
148
views
Only finitely many fundamental groups in $M(n,k,v,D)$?
Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
- 689
-3
votes
1
answer
330
views
Loop space of manifold [closed]
Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...
- 189
-4
votes
1
answer
328
views
Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]
Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?
NOTE: PLEASE avoid the ...
- 612