All Questions
Tagged with differential-topology gn.general-topology
131 questions
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}) \...
- 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 ...
- 21
4
votes
1
answer
904
views
Difference of two-sided and oriented [closed]
I always encounter two definitions: two-sided and oriented (hypersurface or submanifold). What is the difference of them? Which one is stronger?
- 633
0
votes
1
answer
554
views
Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or ...
- 4,195
5
votes
2
answers
562
views
Collared boundary of a non-metrizable manifold
For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally ...
- 397
6
votes
1
answer
711
views
Restrictions of a local diffeomorphism
I am wondering if a local diifeomorphism has the following property (prove or disprove):
Let $M,N$ be differentiable manifolds, and $f:M \to N$ be a local diffeomorphism. Suppose $Z$ is a closed ...
- 151
9
votes
3
answers
1k
views
Link of a singularity
I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$.
If we set $x = x_1+ix_2, y = y_1+iy_2, z ...
user56259
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
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
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
8
votes
2
answers
2k
views
Any 3-manifold can be realized as the boundary of a 4-manifold
We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
- 10.5k
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 ...
- 1,255
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 ...
4
votes
1
answer
414
views
Paracompactness of Quotient by Group Action
Suppose $X$ is a metric space with a free group action by a topological group $G$, which is also a metric space, such that $\pi\colon X \to X/G$ is a fiber bundle.
Does the quotient inherit the ...
- 1,174
4
votes
2
answers
468
views
Is a local diffeomorphism with nice boundary values a diffeomorphism?
Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.
The only situation I can image ...
- 1,804
4
votes
2
answers
338
views
Map between manifolds and open dense subsets
Let $X$,$Y$ be compact, connected, smooth manifolds of the same dimension. Suppose you have a surjective smooth map $f : X \rightarrow Y$, such that $|f^{-1}(p) | \leq k$ for all $p \in Y$.
Let $U \...
- 6,995
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
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
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 $R^{n+1}$. ...
- 697
7
votes
1
answer
614
views
Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:
By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ ...
- 356
8
votes
2
answers
793
views
Does there exist a Haken manifold where all its incompressible surfaces are non-separating?
Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A ...
- 3,751
30
votes
2
answers
2k
views
Does there exist any non-contractible manifold with fixed point property?
Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
- 3,751
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
10
votes
2
answers
488
views
Is there a notion of "space" such that vector bundles can be understood in this way?
Is there a notion of "space" satisfying the following requirements?
Spaces form (at least) a category; morphisms between spaces are called "continuous maps."
Every topological space is a space, and ...
- 3,793
16
votes
4
answers
2k
views
Self-covering spaces
Let $M$ be a connected Hausdorff second countable topological space. I will call $M$ self-covering if it is its own $n$-fold cover for some $n>1$. For instance, the circle is its own double cover ...
- 1,959
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
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
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
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
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
6
votes
0
answers
188
views
Quotients of 4-sphere by smooth $Z_p$ actions with knotted fixed point sets
This question is closely related to another I asked today.
Giffen showed in 1966 that the generalized Smith conjecture is false by constructing for odd $p$ a smooth $Z_p$ action on $S^4$ with fixed-...
- 2,851
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 $...
- 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 ...
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 $...
- 491
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
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
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
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
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
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
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}^\...
- 191
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
-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
7
votes
4
answers
10k
views
Studying topology: which first, algebraic or differential? [closed]
I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...
- 129
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
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^\...
- 81
9
votes
2
answers
647
views
Is the strong Whitney topology connected?
$\def\bbR{\mathbb R}\def\ssp{\kern.4mm}$Put more precisely, let $C$ be the set of all continuous functions $f:\bbR\to\bbR$ when
$\bbR$ has its standard order topology. Let $\mathscr T$ be the set of ...
- 3,584