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
0
votes
1 answer

Homology of the 3-torus

I've been learning about Morse homology, and I find it easy to compute the homology group of surfaces embedded in R3 by defining a Morse function on it, seeking the critical points and the stable und unstable manifolds etc, but what if I am to…
janTschän
0
votes
2 answers

Find the critical points of the function.

Let $M = \{(x, y, z, w) \in \mathbb{R}^4 \ | \ x^4 + y^4 + z^2 + w^2 = 1\}$ and let $f:M \rightarrow \mathbb{R}$ be given by $f(x, y, z, w) = x^3 - z.$ Then it is clear (I have already proven that is) that $M$ is a manifold. What I am having…
dunkindonuts
  • 141
0
votes
1 answer

$f: X \to Y$ is a submersion, $X$ is compact and $Y$ is connected - why $f(X)$ is open?

$f: X \to Y$ is a submersion, $X$ is compact and $Y$ is connected - why $f(X)$ is open? Assume I have proved that for an open set $U \subset X$, $f(U)$ is open. Thank you.
WishingFish
  • 2,412
0
votes
1 answer

When $\operatorname{dim}X = \operatorname{dim} Y$, immersions are the same as local diffeomorphism.

When $\operatorname{dim}X = \operatorname{dim} Y$, show that immersions $f: X \rightarrow Y$ are the same as local diffeomorphism. If $\operatorname{dim}X = \operatorname{dim} Y$, then $\operatorname{dim}T(X) = \operatorname{dim} T(Y)$. Hence,…
1LiterTears
  • 4,572
0
votes
1 answer

Framed manifolds question

Let a $\pi$-manifold be a manifold with the property that its normal bundle is trivial if it is embedded into $\mathbb R^n$ for large enough $n$. Homotopy spheres are $\pi$-manifolds. Here it is stated that the only spheres that admit a framing are…
snailspace
  • 857
0
votes
1 answer

An exercise on Regular Value Theorem

I got really stuck here for problem 2.3.8 on GP: Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and let $K \subset \mathbb{R}^n$ be compact. Show that for any $\epsilon > 0$ there exists $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$…
1LiterTears
  • 4,572
0
votes
0 answers

A direct application of Sard's theorem

The question is let $f: X \rightarrow \mathbb{R}^2$, show that for almost every $c \in \mathbb{R}$, we have that $f^{-1}(\{c\}\times\mathbb{R})$ is a smooth submanifold of $X$. I want to apply Sard's theorem. But the precise statement I plan to use…
1LiterTears
  • 4,572
0
votes
1 answer

Show that $dF_x$ is surjective for all $x$

I am trying to tackle question 2.3.8 on GP, but I haven't figure out the following question yet. Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ for $A \in \mathrm{Mat}_{m\times n}$. Define $F:…
1LiterTears
  • 4,572
0
votes
0 answers

Does there exists a smooth one-one map from $S^2 \to S^1$

While I was able to prove that there is no surjective, smooth map $f:S^1 \to S^1 \times S^1 \times S^1$ but I am unable to prove this. Suppose such a map exists, then by sard's theorem, it has a regular value. Hence there exists a unique $x \in S^2$…
permutation_matrix
  • 1,356
0
votes
1 answer

Volum of the covering of $\bar{S} \geq S$?

The proposition on GP Page 203 says: Let $S$ be a rectangular solid and $S_1, S_2, \ldots$ a covering of its closure of $\bar{S}$ by other solids. Then $\sum$vol$(S_j) \geq$ vol($S$). This does not quite make sense to be - why can we assert that…
1LiterTears
  • 4,572
0
votes
1 answer

Milnor's proof of Sard's theorem

What does the notation mean in the last step of milnor's proof for Sard's theorem $f(x+h) = f(x) + R(x+h)$ where $||R(x+h)|| \leq c||h||^{k+1}$ What do the lines || mean here?
SteveRodgers43
  • 217
0
votes
1 answer

regular value of homotopy F is regular value for F_t

Let $M,N$ be smooth manifolds of the same dimension, let $I=[0,1]$ and say we have a smooth map $F:M \times I \to N$ with regular value $y \in N$. Is $y$ then also a regular value of $F_t := F(\cdot,t)$ for all $t \in I$?
gogoog
  • 13
0
votes
1 answer

0-manifold - final step

$f$ is a Lefschetz map on a compact manifold X. And I need to show the Lefschetz fixed point is isolated. I proved that the graph of f is transversal to the diagonal inside $X \times X$, then I don't know how to proceed from here. Thank you very…
1LiterTears
  • 4,572
0
votes
2 answers

Approximating continuous functions $S^n \to S^n$

I'm trying to check that every continuous function $f:S^n \to S^n$ can be approximated by differentiable ones. Well, by Stone-Weierstrass I can approximate the coordinate functions $f_i:S^n \to \Bbb R$ by differentiable ${\tilde f_i}:S^n \to \Bbb…
Vinicius M.
  • 1,184
0
votes
1 answer

A basic proof on Morse Function

The questions is to show if $f_t$ is a homotopic family of functions on $R^k$, show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for sufficiently small t. I get the previous question leads to the proof,…
1LiterTears
  • 4,572
1 2 3
…
40 41