Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

0
votes
0answers
18 views

State a Product rule giving the formula for h'(a), and prove this formula

Here is the real analysis problem I am trying to solve: Let $\Omega$ be an open subset of $\mathbf{R}$^p. Let f,g: $\Omega$ $\rightarrow$ $\mathbf{R}$ be functions differentiable at a $\epsilon$ $\...
0
votes
0answers
19 views

Generic maps and double points

What is the definition of a generic map between two smooth manifolds? Are all continuous maps between smooth manifolds homotopic to generic ones? Let $N^n, M^{2n}$ be compact smooth manifolds. Why ...
2
votes
0answers
25 views

What is really being asked by “Prove that $S^1 ⊂ R^2$ is a sub manifold”?

I'm self-studying smooth manifolds, and there is some terminology that bothers me a lot. In a lot of books, or homework questions that I looked, there are statements such as Prove that $S^1 ⊂ R^2$...
2
votes
1answer
53 views

Do the two manifolds intersect at a submanifold

Suppose now I have two smooth manifolds of dimension $n-1$, which are given by the zero level sets of two polynomials. Specifically, suppose $M$ is the manifold given by the zero set of the polynomial ...
4
votes
0answers
57 views

Find $f$ such that $f^{-1}(\lbrace0\rbrace)$ is a knotted curve

I would like to solve the following problem (it comes from Morris W. Hirsh, Differential Topology, it's exercise 6 section 4 chapter 1): Show that there is a $C^\infty$ map $f:D^3\to D^2$ with $0\...
0
votes
0answers
20 views

Boundary of a given chain

1) To prepare learning about integration on manifolds, I am trying to find the boundary of a chain given as follows: $\sigma(t_1, t_2) = (1, t_1, t_2) - (0, t_1, t_2) + (t_1, 0, t_2) + (t_1, t_2, 1) -...
0
votes
0answers
14 views

Understanding the idea of a pseudo-gradient vector field

I have the following definition of a pseudo-gradient vector field: Let $V$ be a Banachspace, $E\in C^1(V)$, $\tilde V = \{u\in V \mid DE(u)\neq 0\}$. Then $v: \tilde V \to V$ is called a p.g.v.f. of ...
2
votes
1answer
27 views

A description of the map between Grassmanians $G_1^k \rightarrow G_k$,

We know that $G_k:=co\lim G_k(\Bbb C^n)$ is the classifying space for $k$ dimensional complex vector bundles. With total space $E_k = \{(x,v) \, :|, x \in G_k, v \in \Bbb C ^\infty \}$. So we may ...
1
vote
1answer
41 views

Is the normal bundle construction idempotent?

Let $X$ be a submanifold of $M$. Inductively, let $N_0$ be the normal bundle of $X$ in $M$, and $N_{k+1}$ the normal bundle of $X$ in $N_k$. (Identify $X$ with the zero section of $N_k$, of course.) ...
1
vote
0answers
18 views

Local extendability of structure functor of A-germs of embeddings on a differential manifold

I am reading "Differential Topology" by Morris W. Hirsch. In book some form of Whitney embedding theorem is almost proved (there are details left to verify): every differential manifold $M$ (...
0
votes
1answer
20 views

Are real linear maps of smooth sections locally determined?

Let $ \pi_{ 1 } \colon E_{ 1 } \to M $ and $ \pi_{ 2 } \colon E_{ 2 } \to M $ be smooth vector bundles (of finite rank) over a smooth manifold $ M $, and consider a map $ T \colon \Gamma ( E_{ 1 } ) \...
1
vote
0answers
27 views

Tangent Map of an Isometry and the Shape Operator

I have spent a lot of time on this problem and would appreciate some help. Please bear with me. Let $M$ be a surface and $F\colon\mathbb{R}^3\rightarrow\mathbb{R}^3$ an isometry. Denote with $F_\...
0
votes
1answer
53 views

Connected boundary implies $\pi_1(M,\partial M)=0$.

I have two questions: Let $M$ be a compact connected manifold with boundary. 1, If the boundary $\partial\tilde{M} $ of universal covering $\tilde{M}$ is connected, is $\partial M$ connected? How ...
1
vote
0answers
22 views

Tangent vectors as equivalence classes of triples and ordinary vectors

I am using this document as a reference on tangent spaces etc. In the section on tangent spaces, the author provides three equivalent definitions of a tangent vector, the first being the intuitive ...
5
votes
1answer
47 views

Stiefel-Whitney numbers of manifolds that are boundaries of non-smoothable manifolds

Can a smooth compact manifold be the boundary of a non-smoothable manifold? If so can any of its Stiefel-Whitney numbers be non-zero? Thom's theorem says that a compact smooth manifold has zero ...
0
votes
1answer
29 views

Evaluate the index of a vector field at an isolated zero, using the Brouwer degree

I want to find the index of v(x,y)=(x,-y) at (0,0), which is equal to the degree of the mapping $f:S^1\to S^1$ defined by $$f(x,y)=\frac{v(x,y)}{||v(x,y)||}.$$ To find $\deg(f)$, one needs to pick a ...
0
votes
1answer
36 views

Understanding the meaning of immersion for Manifolds

Let $f: X \rightarrow Y$ be a smooth map of manifolds. $f$ is an immersion at $x \in X$ if $df_x: T_x(X) \rightarrow T_y(Y)$ is an injective map where $y = f(x).$ $T_x(X)$ is the tangent plane of $X$ ...
1
vote
1answer
29 views

Stiefel-Whitney numbers of exotic differentiable structures

The same topological manifold given two distinct differentiable structures comprises two different smooth manifolds. Do these two smooth manifolds have the same Stiefel-Whitney numbers? In other words,...
1
vote
0answers
27 views

Local diffeomorphism of Euclidean space is a global diffeomorphism

I am learning about the Legendre transform and found in Mac Lane's Geometrical Mechanics lecture notes (v1 p.54) the following inversion theorem: The Legendre transformation $\ v\in V\mapsto dL(v)\...
1
vote
0answers
24 views

Second countability of Minkowski double cones.

In the 4-dimensional Minkowski spacetime, for a given point $x=(x^0,x^1,x^2,x^3)$, its timelike future or past set is defined as, $I^{\pm}(x)= \{y=(y^0,y^1,y^2,y^3) \in \mathbb{R}^4: \eta_{\mu \nu}(y−...
1
vote
1answer
16 views

Determining the linear independence of tangent vectors at a point on the manifold

We define the tangent space at a point, say $x_0$, on the manifold $M$ as the set of all derivations, i.e maps which maps smooth maps from a neighbourhood of $x_0$ to real numbers to real numbers. ...
0
votes
0answers
12 views

Given a smooth map $\sigma$, and a linear isomorphism $K$, is there a smooth map $\tau$ s.t $D(\tau \circ \sigma^{-1}) = K$

Let $(M, \Sigma)$ be a smooth manifold, and $\sigma, \tau$ be two smooth charts defined on the neighbourhood of $x_0 \in M$. Then by definition $$\tau \circ \sigma^{-1}$$ is a diffeomorphism from an ...
1
vote
1answer
73 views

A map $f:S^n\to S^n$ of odd degree must carry some pair of antipodal points into a pair of antipodal points

I want to prove by contradiction and hence assume that $f$ does NOT carry any pair of antipodal points into a pair of antipodal points. A smooth homotopy $F(x,t):S^n\times[0,1]\to S^n$ is well defined ...
0
votes
0answers
32 views

Extending solution to a differential equation.

I have to find $\hspace{1ex}$ $h:\mathbb{R}\rightarrow \mathbb{R}^{n}$ $\hspace{1ex}$ satisfying $\hspace{1ex}$ $$\vec g(x)=\frac{ h'(x)}{\sqrt{1-\langle h(x),h(x)\rangle}}$$where $g:\mathbb{R}\...
0
votes
2answers
54 views

Why is the image of the map $f(x) =(sin(x),1- cos(x))$ for $x\in [0, 2\pi)$ and $f =\operatorname{id}$ otherwise not a submanifold of $\mathbb{R}^2$?

Consider the map $f: (-\infty, 2\pi) \to \mathbb{R}^2$ s.t $f(x) = (sin(x), 1- cos(x))$ for $x\in [0, 2\pi)$ and $f(x) = (\operatorname{id}(x), 0)$ for the rest of its domain. In the book of ...
4
votes
0answers
51 views

Topological invariance of compactly supported de Rham cohomology

It is well-known that if we are given two smooth manifolds (without) boundary, whose underlying topological spaces are homotopic, then the de Rham cohomologies $H^k_{dR}$ of $M$ and $N$ are isomorphic ...
0
votes
1answer
44 views

Left translation on Lie group of a discrete subgroup is properly discontinuous

This question has been asked before here and there but has not received answers which make clear my difficulties understanding this argument. I am quite rusty in both group theory and topology, and I ...
1
vote
1answer
58 views

Prerequisites to understand Immersion conjecture proof by Cohen (1985)

I would like to understand the proof by Cohen of the Immersion Conjecture, but since it is a relatively recent work I probably need lots of prerequisites: my background is composed by an introduction ...
0
votes
0answers
14 views

continuity of isomorphism of unit circle

I try to show $\mathbb{S}^1\cong[0,1)$, by the map $f(x) = (\cos2\pi x,\sin2\pi x)$, for $x\in[0,1)$. It's clear that $f$ is continuous and bijective. But I don't know how to show the inverse map $f^{-...
1
vote
0answers
15 views

Principal fibre bundles with constant transition functions

I guess this is not limited to principal bundles, however those are my primary interest in asking this question. Let $(P,\pi,M,G)$ be a principal fibre bundle over $M$ with structure group $G$. ...
0
votes
0answers
7 views

Understanding Submersion for kernel density estimation

The kernel density estimation, defines a function $$ P(\mathbf x, \sigma) = \frac 1 N \cdot \sum_{i=1}^N K_i(x, \sigma) $$ with (here) $K$ being the (multivariate) Gaussian function $$ K_i(\mathbf ...
0
votes
0answers
11 views

The Möbius Strip diffeomorphism

Are Möbius Strip and $\mathbb{R}P^2$ diffeomorphic? I really don't know how to approach this, I have tried to construct a diffeomorphism but couldn't get anywhere.
0
votes
0answers
45 views

Are the spaces $T_pM$ and $\mathbb R^n$ homeomorphic?

Let $T_pM$ be the tangent space at a point $p$ in a n-dimensional smooth manifold $M$. In addition, if we assume $(M,g)$ as a smooth Riemannian manifold, then $T_pM$ is a n-dimensional real normed-...
1
vote
0answers
23 views

winding number and Jordan-Schoenflies Theorem

I have a statement which is related to the smooth Jordan-Schoenflies Theorem. I can verify it in some simple cases. The setup is: Let there be a $C^1$ immersion $\gamma : S^1 \to \mathbb{R}^2$. ...
1
vote
1answer
48 views

Map of $\mathbb{R}^3-Knot \to S^1$

Reading Bachman's "A Geometric Approach to Differential Forms", in section 7.8.1 about the Lining Number invariant, I have stumbled upon the following assertion. Let the knot $K$ be defined as a (...
0
votes
0answers
16 views

Structure of level-sets in a bounded domain.

Let $M \subset \mathbb{R}^N$, be an open set where $M$ contains the closed set $E = \{x \in \mathbb{R}^N | x_i \in [0,1], i = 1,\dots,N \}$. Consider a function $f: M \to \mathbb{R}$ which is smooth (...
2
votes
3answers
61 views

Proving that $\{ x \in \mathbb{R}^n : |x| = 1, x\geq 0\}$ is homeomorphic to $\overline{B_1(0)} \subset \mathbb{R}^{n-1}$

I want to show that $A= \{ x \in \mathbb{R}^n : |x| = 1, x\geq 0\}$ is homeomorphic to $\overline{B_1(0)} \subset \mathbb{R}^{n-1}$, where $\overline{B_1(0)}$ is the closed unit ball. I was thinking ...
1
vote
1answer
13 views

Rank of a map restricted to the boundary of a manifold

Let $M$ be a smooth manifold with boundary, $N$ a smooth manifold and $F : M \to N$ smooth. Define $$f = F|_{\partial M}.$$ If $df_x$ has rank $r$ at $x \in \partial M$, can we conclude that the rank ...
1
vote
1answer
32 views

Complement of tubular neighborhood

Let $M$ be a closed, connected, orientable and embedded surface inside the unit 3-sphere $\mathbb{S}^3$ and consider a small tubular neighborhood $U$ of $M$: $$U = \{ x \in \mathbb{S}^3 : d(x, M) \...
1
vote
0answers
110 views

Can this integral be made non-positive?

Let $M \subset \mathbb{S}^3$ be a closed, connected and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 ...
5
votes
2answers
103 views

Linking number definitions: equivalent or not?

The linking number $\text{link}(C_1,C_2)$ of two disjoint smoothly embedded oriented circles $C_1,C_2$ in $\Bbb{R}^3$ has several equivalent definitions, one of which is the degree of the map $S^1\...
3
votes
0answers
23 views

Manifold with a good exhaustion [duplicate]

Let $M$ a smooth manifold such that $M=\bigcup_{i=1}^{\infty} U_i$ for $U_i \subset U_{i+1}$, where $U_i$ is an open set of $M$ which is diffeomorphic to $\mathbb R^n$. Can we prove that $M$ is ...
0
votes
0answers
23 views

Proving the Ham-Sandwich Theorem

A version of the "Ham-Sandwich theorem" states that if we have $n$ finite Borel measures $\nu_1 , \dots, \nu_n$ on $\mathbb{R}^n$ which assign zero measure to hyperplanes then there exists a halfspace ...
2
votes
1answer
69 views

Is there an orientable $3$-manifold with non-vanishing $w_2$?

In the case that $M$ is a closed orientable $3$-manifold, using Wu's formula we can show $w_1(M) =0 \implies w_2(M) =0$, and so $w_3 = w_1w_2 + Sq^1 w_2 = 0$ (or you can use the fact that $\chi(M)=0$ ...
1
vote
0answers
23 views

Jet bundle cohomology

Consider a base manifold $\mathcal{M}$ and a smooth bundle $E\to\mathcal{M}$. I am interested in the cohomology groups of the variational bicomplex associated with the jet bundle $J^{\infty}E$. In ...
5
votes
1answer
64 views

Existence of $S^1$-action on a vector bundle and computing its characteristic classes

The existence of an $S^1$ action sometimes helps us in computing topological invariants. For example we can compute the Euler characteristic looking at the fixed point set (see Euler characteristic ...
2
votes
1answer
55 views

Are the 1-parameters subgroups of $SO(3)$ closed?

I'm trying to solve the following question Question: Prove that all $1$-parameters subgroup of $SO(3)$ are closed. Does this statement holds for $SO(n),$ $n>3$? Some comments The $1$-...
0
votes
0answers
22 views

How is $G_1 = \cup_{k=1}^{\infty} U^k$ a closed subgroup of $G^0$? for Lie group $G$

Let $G$ be a Lie group, and $G^0$ the identity component. Let $U$ be an open subset of $G^0$ containing the identity such that $U = U^{-1}$. Let $$ G_1 = \cup_{k=1}^{\infty} U^k. $$ Clearly $G_1$ ...
6
votes
2answers
126 views

Let $M$ be a smooth n-manifold($n\geq 1$).Prove that $M$ admits a diffeomorphism $f : M \to M$ which is not the identity

I don't know how to come up with a proof for this. I think I have to come up with a bump function and do it locally on each chart and then extend it to the whole manifold?? May be I am wrong.. Thanks ...
1
vote
0answers
37 views

smooth singular vs.singular homology

In the GTM218,Theorem 18.7: For any smooth manifold $M$,the map $l_*:H_p^{\infty}(M)\rightarrow H_p(M)$ induced by inclusion is an isomorphism. Is it also true for smooth manifold with boundary?(It ...