Mathematics Stack Exchange
Mathematics 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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

7287 questions
2
votes
1 answer

Why the tangent bundle of a smooth manifold is an oriented manifold?

I need help with the following question. I am not sure how to begin. Any help will be appreciated. Thank you! For any smooth manifold $M,$ the tangent bundle $TM$ is an oriented manifold.
Susan
  • 1,205
2
votes
1 answer

Differential Topology Question on Complex Projective Space

This question seems like it would be very hard to do directly. I wouldn't know where to begin. I was wondering if anyone had a very slick proof of this. The only thing I think is easy is that its connected. The rest seems like the only way I…
Susan
  • 1,205
2
votes
0 answers

Question Concerning the classification of 1-manifolds

I am having trouble proving the following statement used in proving the classification of 1-manifolds. Any help would be great. Thank you. Let $L$ be a subset of $X$ diffeomorphic to an open interval in $\mathbb{R}$, where dim $X=1.$ Then the…
Susan
  • 1,205
2
votes
1 answer

Using a submersion and a non-vanishing vector field on the target to obtain a non vanishing vector on the source.

I was thinking about the following problem which says: Let $f: M \rightarrow N$ be a smooth submersion. Let $W$ be a nowhere vanishing vector field on $N$. Construct a nowhere vanishing vector field $V$ on $M$ such that $f_{*}V=W$. I thought about…
user135520
  • 2,137
2
votes
0 answers

Can assume $f$ agrees with a vector bundle map?

I am reading chapter 7 lemma 2.2 in "Differential topology" by M.Hirsch which is about Thom's isomorphism. The lemma goes as Lemma 2.2: Let $Q$ be a compact manifold, and $B$ be a compact manifold without boundary. Then every map $f:Q\to E^*$ is…
Y.Guo
  • 754
2
votes
0 answers

What is $S^1 \times S^3$ with an open disk removed?

I am trying to calculate the de Rham cohomology of the connected sum of $S^1 \times S^3$ with $\mathbb CP^2$. But I have trouble in figuring out what is $S^1 \times S^3$ with a disk removed. If I have that figured out, I can proceed using Mayer…
penny
  • 201
2
votes
1 answer

Brouwer's degree: equivalent definitions

I am reading "Topological degree theory and applications" by O'Regan, Cho and Chen. I am stuck on the start: Consider $\Omega\subset \mathbb{R}^n$ open and bounded and let $f\in C^1(\bar \Omega)$, if $p \not \in f(\partial\Omega) $ and $J_f(p)\not…
Moritzplatz
  • 1,521
  • 14
  • 28
2
votes
0 answers

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 $M$ (paracompact, Hausdorff, with countable base) of dimension…
Shingle
  • 529
2
votes
2 answers

Composition of smooth maps

From section 1, problem 3 of Differential Topology by Guillemin and Pollack: Let $X \subset R^N, Y \subset R^M, Z \subset R^L$ be arbitrary subsets, and let $f : X \to Y, g : Y \to Z$ be smooth maps. Then [show that] the composite $g \circ f : X…
Red
  • 21
2
votes
0 answers

A differentiable manifold class $C^k$ but not class $C^{k+1}$

Please show that the graph of $f(x)=|x|^λ$, where $k<λ
Adele
  • 99
2
votes
1 answer

Doubt on proof: Orientation preserving diffeomorphism is isotopic to identity

I am reading Milnor's ''Topology from the differentiable viewpoint'' and in the chapter about vector fields there is a Lemma that states that given $f:\mathbb{R}^n \to \mathbb{R}^n$ an orientation preserving diffeomorphism, then $f$ is isotopic to…
Bajo Fondo
  • 1,099
  • 7
  • 17
2
votes
0 answers

Normal bundle to $Z$ in $Y$ is a manifold with same dimension as $Y$ (Exercise 2.3.12 of Guillemin-Pollack)

Here we have $l > k$. I let $\phi : U \times \mathbb R^l \longrightarrow N(Z;\mathbb R^M)$ be the associated parametrization. I thought that to restrict $U \times \mathbb R^l$ to $U \times \mathbb R^k$ one should consider the projection map $\pi :…
Cookie
  • 13,532
2
votes
0 answers

Proving that for a $C^k$ function, the measure of the set of singular values is $0$

Let $f:R^m\to R^n$ be a $C^k$ function, where $k>\max \{0,,m-n\}$. Then the lebesgue measure of the set of singular values is $0$. I've been trying to prove this. I came up with the following argument: let $n=1$, and $m>1$. Then the set of…
user67803
2
votes
1 answer

If $M$ is a manifold, such that $\dim(M)>1$, then we can find 2 disjointed paths.

I think this is easy, but I'm very stuck in this problem. Question: Let $M$ be a connected smooth manifold without boundary with $\dim M >1$, and $a_1,a_2,b_1,b_2$ $\in M$ are different points. Then there are two paths $\gamma_1,\gamma_2:[0,1]\to…
Matheus Manzatto
  • 3,231
2
votes
1 answer

What are the "standard coordinate functions" on $R^n$?

I'm reading page 44 of Differential Topology by Guillemin. I am trying to understand this proof: Suppose that x is any point in X (a manifold sitting in $R^n$. and that $x_1,....x_n$ are the standard coordinate functions on $R^n$ (So...is just the…
Ecotistician
  • 1,086
1 2 3
…
40 41