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8 votes
0 answers
229 views
+300

Maps with small fibers between manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold. Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
Matthew Kvalheim's user avatar
-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 ...
Bastam Tajik's user avatar
2 votes
0 answers
136 views

Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures "Global Conjecture: There is a dense set $D$ of dynamics such that any element of ...
NicAG's user avatar
  • 247
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 ...
NicAG's user avatar
  • 247
29 votes
2 answers
2k views

Contractibility of the space of Jordan curves

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$. If the curves are ...
Mohammad Ghomi's user avatar
2 votes
1 answer
119 views

Density of smooth bi-Lipschitz maps in smooth maps

Setup/Motivation: Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
Carlos_Petterson's user avatar
2 votes
1 answer
130 views

Gluing isotopic smoothings

Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
UVIR's user avatar
  • 803
6 votes
0 answers
136 views

A particular case of the general converse to the preimage (submanifold) theorem

I was thinking whether it would be possible to develop a converse to the preimage theorem in differential topology and I found the following post: When is a submanifold of $\mathbf R^n$ given by ...
geooranalysis's user avatar
8 votes
0 answers
198 views

A modified version of the converse to the Sard's Theorem

When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
pureorapplied's user avatar
6 votes
0 answers
297 views

Regarding homology of fiber bundle

Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
tota's user avatar
  • 585
1 vote
1 answer
291 views

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 ...
MathDG's user avatar
  • 272
7 votes
2 answers
562 views

Is the union of a compact and the relatively compact components of its complementary in a manifold compact?

I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is ...
Saúl RM's user avatar
  • 10.6k
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 $M \subset \mathbb{R}^n$. $D_r (M): = \bigcup\limits_{x \in M} D_r (x)$ $Int_r (M): = \{x ~|~ D_r(x) \subset M\}$ ...
Arshak Aivazian's user avatar
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^...
Dmitrii Korshunov's user avatar
2 votes
1 answer
560 views

Collar neighborhood theorem for manifold with corners

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
301 views

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 with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
ABIM's user avatar
  • 5,405
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,...
Francine Laporte's user avatar
5 votes
2 answers
308 views

Is this subset of matrices contractible inside the space of non-conformal matrices?

Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and $\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...
Asaf Shachar's user avatar
  • 6,741
2 votes
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 ...
Senthan Sara's user avatar
3 votes
0 answers
64 views

Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following: Is $X$ ...
user44172's user avatar
  • 541
3 votes
1 answer
178 views

Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...
user avatar
12 votes
2 answers
778 views

Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold. a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ? b) If not, are there some concrete ...
H1ghfiv3's user avatar
  • 1,255
7 votes
1 answer
545 views

Could we always find a line to intersect transversally with a given compact manifold?

This problem may be an embarrassing one, but I could not prove it even for the $1$ dimensional case. Here is the problem: Question 1. $M$ is a compact $n$-dimensional smooth manifold in $R^{n+1}$. ...
Hu xiyu's user avatar
  • 697
2 votes
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 ...
Dean Barber's user avatar
3 votes
0 answers
83 views

Is the increasing union of disk bundles a disk bundle?

Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $...
Matthew Kvalheim's user avatar
7 votes
1 answer
692 views

Homotopically trivial vs isotopically trivial diffeomorphisms

Let $M$ be a manifold. Let's say $M$ is smooth, connected, oriented. We can also assume that $M$ is closed if that makes things easier. Let $\mathit{Diff}(M)$ denote the group of diffeomorphisms of $...
seub's user avatar
  • 1,347
4 votes
2 answers
619 views

Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
Shinichiro Nakamura's user avatar
23 votes
1 answer
2k views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...
Matthew Kvalheim's user avatar
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 ...
Blake's user avatar
  • 109
3 votes
1 answer
270 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
Ben's user avatar
  • 35
2 votes
1 answer
1k views

Is it true that given any two point in $M$ if there exists an unique geodesic joining those two points, then $M \sim \mathbb{R^n}$ [closed]

This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points. DOUBT: Suppose $M\subset \mathbb{...
Anubhav Mukherjee's user avatar
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}^\...
Kathrin L.'s user avatar
2 votes
1 answer
1k views

Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all: Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...
Kathrin L.'s user avatar
-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 ...
MyIsmail's user avatar
  • 189
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 ...
Kathrin L.'s user avatar
8 votes
2 answers
2k views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\...
Kathrin L.'s user avatar
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 ...
Paul's user avatar
  • 914
3 votes
3 answers
351 views

Diffeomorphism with prescribed behaviour

If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...
Alex M.'s user avatar
  • 5,407
2 votes
1 answer
391 views

(n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper http://www.unige.ch/~...
A random mathematician's user avatar
4 votes
2 answers
414 views

Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and $S \subset M $ a topological subspace of $M$. We say that $S$ has "dimenion" at most $k$ if $S$ is a subset of $$ X_1 \cup X_2 \ldots ...
Ritwik's user avatar
  • 3,245
12 votes
0 answers
460 views

3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
Murat Saglam's user avatar
2 votes
1 answer
407 views

Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint n-...
Hugo Chapdelaine's user avatar
113 votes
4 answers
13k views

Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
Daniel Moskovich's user avatar
33 votes
4 answers
7k views

Topology of function spaces?

Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds. Let $C^\infty(X,...
José Figueroa-O'Farrill's user avatar
14 votes
2 answers
1k views

Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$?

Question: Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? I'd guess that the answer doesn't depend on choice of differentiable structure, but maybe it does. Motivation:...
Zack's user avatar
  • 787
16 votes
2 answers
2k views

Compactification of a manifold

This is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when I say minimal I think something like adding a finite ...
Italo's user avatar
  • 1,727