All Questions
Tagged with differential-topology gn.general-topology
131 questions
4
votes
1
answer
121
views
Condition for existence of a continuous function realizing a partition
Let $\{U_i\}_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup_{i =1 }^{ I} \overline{U_i}$. ...
4
votes
2
answers
273
views
Question about lower homology class of cobordism
Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...
- 703
4
votes
0
answers
97
views
Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?
Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?
The assumption that $ N_G(H)/H $ is finite cannot be weakened ...
- 4,081
4
votes
0
answers
155
views
Equivalence of definitions of Hirsch and Wall of strong $C^r$-topologies
I've been reading about strong (and weak) $C^r$-topologies on the space of $C^r$-maps between $C^s$-manifolds $M$ and $N$ ($s \ge r$) from the textbooks of Hirsch and Wall (both called Differential ...
- 281
4
votes
0
answers
2k
views
A closed set which is the closure of its interior points
This is a cross-post to MSE to this question
https://math.stackexchange.com/questions/3267497/a-set-which-is-the-closure-of-its-interior-points
There is a related question in MO:
Closure of the ...
- 293
4
votes
0
answers
452
views
Finite good covers on smooth manifolds
Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex.
Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...
- 180
4
votes
0
answers
133
views
Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber
Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...
- 404
3
votes
2
answers
410
views
A question on continuous maps from Möbius to itself
Let $M$ denotes the Möbius strip. Then is it true that
For every continuous map $f:M\to M$ there is $x\in M^\circ$ ($x\notin\partial M$) such that $f(f(x))=x$?
- 4,195
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 ...
- 5,407
3
votes
2
answers
472
views
Regularity of lipschitz and derivable function
Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
- 4,074
3
votes
2
answers
306
views
Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$
$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$.
I would like ...
- 2,533
3
votes
1
answer
351
views
How to define relative orientation in terms of (co)homology?
Let $f\colon X\to Y$ be a smooth surjective map of smooth manifolds of dimension $n$ which are not necessarily orientable. A relative orientation of $X$ over $Y$ consists of an isomorphism $\psi\colon ...
- 3,031
3
votes
2
answers
601
views
SU(2) representations of alternating knot groups
Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
- 1,129
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 ...
- 35
3
votes
1
answer
99
views
cartesian product rigidity for the punctured open disc
Q1: Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to $S^{...
- 7,609
3
votes
2
answers
300
views
Discriminant locus in knot space
Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$.
The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".
Let $f$ be a knot with $n$ double ...
- 5,063
3
votes
1
answer
419
views
Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition
Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the ...
user17778
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 $...
user133532
3
votes
0
answers
159
views
A question regarding weak Whitney embedding theorem
The weak Whitney embedding theorem states that any continuous map $f: D^n \to \mathbb{R}^{2n+1}$ (Let us focus on $D^n$ for this question) can be approximated (in $C^0$-norm) by embeddings. A counter ...
- 31
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 $X$ ...
- 541
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 $...
- 491
3
votes
0
answers
115
views
Characterization of global sections (which are not products) of a sheaf which is locally a product
In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
- 397
2
votes
1
answer
308
views
Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)
every smooth manifold can be triangulated, is it true for orbifold? Is it a known result? If yes, is there any reference?
reply to the comment : G does not need to be any subgroup of Sn , any ...
- 21
2
votes
1
answer
178
views
Differentials in different topologies
I have read (In French)that the differential of a function depends on the topology and not the norm, the latter is rather easy to grasp, the first is hard for me to construct.
Norms being equivalent ...
- 375
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{...
- 3,751
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/~...
2
votes
1
answer
212
views
A definition of linking number for knots in $S^3$ using chains in $D^4$
I meet this problem when reading Rolfsen's Knots&Links. After giving 8 different definitions of linking number for knots in $S^3$, he left an exercise: Given disjoint PL knots $J$ and $K$ in $S^3=\...
- 225
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 ...
- 5,405
2
votes
1
answer
253
views
What are some surprising facts that happen after you remove a point to a space? [closed]
There are some facts that are really impressive after you remove a point to a space. Some typical examples are the existence of exotic spheres or the fact that
$S^4$ is not almost complex. Or some not ...
2
votes
1
answer
166
views
What is the most symmetric configuration of four 2-surfaces linked in $S^4$?
What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$?
To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
- 10.5k
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 ...
- 191
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 ...
- 195
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 ...
- 5,405
2
votes
1
answer
163
views
Homological restrictions on certain $4$-manifolds
I am not very familiar with the non-compact $4$-manifold theory. So I apologize if the following question is very silly.
Let $X$ be a non-compact, orientable $4$ manifold that is homotopic to an ...
- 1,177
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 ...
- 803
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 ...
- 5,405
2
votes
1
answer
89
views
Constructivity of zeros demanded by topological degree
Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...
- 5,827
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-...
- 7,609
2
votes
1
answer
512
views
Question about analytic curves
Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...
- 203
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 ...
- 247
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
votes
0
answers
86
views
Existence of a proper Morse function
I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy ...
- 1,177
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 ...
- 21
2
votes
0
answers
138
views
Does any smooth oriented closed orbifold have a fundamental class
This thread:triangulation of orbifolds
has shown that any smooth closed orbifold has a triangulation. My further question is: if the difference of any two triangulations $P$ and $Q$ is a boundary of a ...
- 781
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 ...
- 79
2
votes
0
answers
109
views
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
652
views
Surjectivity of maps between spheres [closed]
I am wondering how to prove that a non-zero degree map from $S^n \to S^n$ is surjective. For example, identifying $S^1 \subset \mathbb{C}$, we can take $f:S^1 \to S^1$ via $f(z) = z^k$ with $k\neq 0$. ...
- 21
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
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
1
vote
3
answers
660
views
How can I construct a closed manifold from a finite CW complex?
If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open ...
- 1,177