All Questions
Tagged with gn.general-topology differential-topology
131 questions
140
votes
7
answers
34k
views
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
- 151k
112
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 ...
- 22.1k
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
36
votes
2
answers
3k
views
Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
- 22.8k
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,...
- 33.3k
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
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 ...
- 7,149
25
votes
1
answer
1k
views
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
- 439
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 $...
- 781
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
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
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 ...
- 1,727
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:...
- 787
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
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
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 $S^2 \times M_2$?
- 131
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
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
9
votes
2
answers
646
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
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
8
votes
1
answer
708
views
Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$
I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow \...
- 2,696
8
votes
2
answers
792
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
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
8
votes
1
answer
584
views
The closure of the set of injective continuous functions
Setup/Notation:
Let $n,m\in \mathbb{N}$ and let $C(\mathbb{R}^n,\mathbb{R}^m)$ be the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ equipped with the compact-open topology. Let $...
- 5,405
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 (...
- 349
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
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
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
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 ...
- 10.6k
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
0
answers
185
views
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 ...
- 781
6
votes
1
answer
815
views
When is a Topological pushout also a Smooth pushout?
I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
- 732
6
votes
1
answer
709
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
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 ...
- 197
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 ...
- 585
6
votes
0
answers
196
views
Logarithm on formal power series continuous?
Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...
- 463
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
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
5
votes
2
answers
292
views
Simultaneously minimizing intersections
This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in $\...
- 13k
5
votes
1
answer
279
views
Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.
...
- 781
5
votes
2
answers
609
views
Exponential rule for Whitney-$\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
5
votes
1
answer
738
views
Characteristic classes of a fibered sum
I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.
Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for simplicity),...
- 732
5
votes
1
answer
379
views
Proving the Cork Theorem
I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...
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
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 ...
4
votes
1
answer
903
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
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
4
votes
2
answers
466
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
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 ...
- 3,245
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