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.

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Change in the topology of $f^{-1}(c)$ as $c$ passes through the critical value.

Pictorially examine the catastrophic change in the topology of $f^{-1}(c)$ as $c$ passes through the critical value, where $$f (x, y, z) = x^2 + y^2 - z^2.$$ I don't have the slightest idea ...
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studying compact $\partial$-$n$-manifolds via closed $n$-manifolds?

What would be counterexamples to the following statement: It is not true that any $n$-manifold with boundary is a $n$-manifold with finitely many embedded disjoint open disks removed, since that ...
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144 views

The inverse of homogenous function

Given homogenous function $p$ with order $m$, how can I show that $$p^{-1}(a) = (\frac{a}{b})^{\frac{1}{m}}p^{-1}(b)?$$ The original question is: Let $p$ be any homogeneous polynomial in ...
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Finite approximation of path space.

Let $M$ be a connected Riemannian manifold, $\Omega(M)=\Omega$ the path space and $E: \Omega \to \mathbb{R}$ the energy function. We can define $\Omega^c:=E^{-1}([0,c])$ and $\Omega(t_0, \dots, t_k)$ ...
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70 views

$\mathbb{R}^k$ and $\mathbb{R}^k$ are trivially diffeomorphic.

Is this claim correct? If so, is it because identity is the diffeomorphism?
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GP 1.4.4 An extension of partial converse of preimage theorem.

This is exercise 1.4.4 on Guillemin and Pollack's Differential Topology Suppose that $Z \subset X \subset Y$ are manifolds, and $z \in Z$. Then there exist independent functions $g_1, \dots, g_l$, ...
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196 views

The independence of gradient.

I am trying to solve this problem: Show that the curve $t \to (t, t^2, t^3)$ embeds $\mathbb{R}^1$ into $\mathbb{R}^3$. Find two independent functions that globally define the image. Are your ...
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1answer
92 views

How to draw the conclusion that $f$ is continuous?

Given $X$ is compact and $Y$ connected, and $f$ is a submersion. How to draw the conclusion that $f$ is continuous? In my book, submersion is defined as:
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243 views

Projection, canonical immersion/submersion - are they equivalent, and are they open maps?

I am very confused with the concept of projection with the introduction of immersion and submersion. By local immersion/submersion theorem, for a simmersion/submersion $f$, there is is a canonical ...
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1answer
61 views

$f: X \to Y$ 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.
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124 views

What is $g$ in Guillemin and Pollack's Differential Topology?

Is it canonical immersion when it appears on Page 15, and cannonical submersion on Page 20? I never really see where it is defined, except for Page 15: "Define $G$ s that $g = G \circ$ (canonical ...
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Non degenerate critical points.

Let $K$ be a compact subset of the Euclidean space $\mathbb{R}^n$; let $U$ be a neighborhood of $K$ and let $f:U \rightarrow \mathbb{R}$ be a smooth function such that all critical points of $f$ in ...
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27 views

The solution set in $\mathbb{R}^3 - \{0\}$ of $x^d +y^d= z^d$ has the form $p^{-1}(X_d)$.

I am wondering if my proof is legit? The ending looks rather soft. I don't know whether it is correct, or how to rephrase it if it is correct. Let $p: \mathbb{R}^3 - \{0\} \to \mathbb{R}P^2$ be ...
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1answer
102 views

How can I show that $a,b \in Z$?

I have a question to the end of my proof for the problem 1.3.10 on Guillemin and Pollack's Differential Topology: Generalizaition of the Inverse Function Theorem: Let $f: X \rightarrow Y$ be a ...
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53 views

submanifold and open subset

If $U$ is an open subset of the manifold $X$, check that $$T_x(U) = T_x(X) \text{ for } x \in U.$$ I only proved when $U$ is an open subset of the manifold $X$, which is not true for ...
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Show that the index of $r$ must be the sum of the indices of $p$ and $q$.

Could someone give me some help to get started with this question? Don't even have the slightest idea.. =( Suppose a vector field v on $\mathbb{R}^n$ has exactly two isolated zeros $p, q$, and $p, ...
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1answer
28 views

Question on 2-chain on $\mathbb{R}^3$

Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that ...
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GP 1.3.9(b) Every manifold is locally expressible as a graph.

This is exercise 1.3.9(b) on Guillemin and Pollack's Differential Topology I believe I am pretty much done with this problem, but I still do not understand why the last step shows the existence, and ...
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140 views

$\mathbb{S}^2$ as a fibre bundle

I know, by the Hopf fibration, that $\mathbb{S}^3$ is an $\mathbb{S}^1$-fibre bundle over $\mathbb{S}^2$. Can $\mathbb{S}^2$ be an $\mathbb{S}^1$-fibre bundle over some manifold $M$?
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Related to GP 1.3.9 - Is projection function smooth?

I start to think of this question when I attempt Ex 1.3.9 on Guillemin and Pollack's Differential Topology GP 1.3.9(b) Every manifold is locally expressible as a graph.. I am under the impression ...
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61 views

Relate to GP 1.3.9 - Differentiating $x_{i_1}, \dots, x_{i_k}$ result span($e_{i_1}, \dots, e_{i_k}$)?

I start to think of this is question when I attempt exercise 1.3.9 on Guillemin and Pollack's Differential Topology Consider the projection $\varphi: \mathbb{R}^N \rightarrow \mathbb{R}^k$: $$(x_1, ...
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66 views

Prove that every point has a neighborhood on which the restrictions of some $k$-coordinate functions form a local coordinate system.

Let $x_1, \dots, x_N$ be the standard coordinate functions on $\mathbb{R}^N$, and let $X$ be a $k$-dimensional submanifold of $\mathbb{R}^N$. Prove that every point $x \in X$ has a neighborhood on ...
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99 views

Isomorphism of vector bundles (exercise 6.2 of Bott, Tu)

I'm self-studying the book by Bott & Tu "Differential forms in algebraic topology" and I'm having problems with exercise 6.2. It says "Show that two vector bundles on $M$ are isomorphic iff their ...
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Restriction to a line is an immersion.

I have proved that $g: \mathbb{R}^1 \rightarrow S^1, g(t) = (\cos 2 \pi t, \sin 2 \pi t)$, is a local diffeomorphism, as well as that $G: \mathbb{R}^2 \rightarrow S^1 \times S^1, G = g \times g$ ...
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1answer
102 views

Every diffeomorphism induces a bundle isomorphism?

I'm starting to learn some vector bundle theory and I have the next question. If I have a diffeomorphism $f:M \rightarrow M$ and $E$ is a vector bundle with base $M$, is it true that there exists a ...
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1answer
66 views

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 ...
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37 views

Restriction to any submanifold of its domain is still immersion.

If $f$ is an immersion, prove its restriction to any submanifold of its domain is an immersion. Consider a submanifold $\tilde{X}$ of $X$, and take any point $p \in \tilde{X}$. Then when ...
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101 views

$d(f \times g)_{x,m} = df_x \times dg_m$?

(a) $d(f \times g)_{x,m} = df_x \times dg_m$? Also, (b) does $d(f \times g)_{x,m}$ carry $\tilde{x} \in T_x X, \tilde{m} \in T_x M$ to the tangent space of $f$ cross the tangent space of ...
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Lifted Diffeomorphism

Suppose to have a diffeomorphism $\phi$ of the d-dimensional torus to itself, and suppose to lift it to a morphism of $\mathbb{R}^d$ to itself. I have proved that is still invertible. How i proof that ...
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608 views

Local diffeomorphism is diffeomorphism provided one-to-one.

For the problem Guillemin & Pallock's Differential Topology 1.3.5, I am not confident with my proof. Prove that a local diffeomorphism $f: X \rightarrow Y$ is actually a diffeomorphism of $X$ ...
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625 views

local diffeomorphism on $\mathbb{R}$ and on manifolds.

I find the proof of diffoemorphism in Guillemin & Pallock's Differential Topology 1.3.3 is more or less independent of the fact that the manifold happen to be $\mathbb{R}$, and therefore are the ...
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If $f$ and $g$ are immersions, show that $f \times g$ is.

Is this proof correct? I am particularly uncertain with the last step. Consider $f: X \to Y, g: M \to N$. $\forall x \in X, df_x: T_x(X) \to T_y(Y)$ is injective. Similarly, ...
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The tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ - Is this proof legit? [duplicate]

If $\triangle$ is the diagonal of $X \times X$, show that its tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ Is the following proof legit? $T_{(x,x)} \Delta ...
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1answer
150 views

The general idea of prove openness.

I never really get the idea of proofs involves openness, here's an example: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a local diffeomorphism. Prove that the image of $f$ is an open interval. ...
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The image of $I$ is an open interval and maps $\mathbb{R}$ diffeomorphically onto $\mathbb{R}$?

This is Problem 3 in Guillemin & Pallock's Differential Topology on Page 18. So that means I just started and am struggling with the beginning. So I would be expecting a less involved proof: ...
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1answer
33 views

Augument, and injectivity.

I am having much trouble reading the proof Local Immersion Theorem in Guillemin & Pallock's Differential Topology on Page 15. Now we try to augment $g$ so that the Inverse Function Theorem may ...
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1answer
50 views

Why the matrix of $dG_0$ is $I_l$.

I am reading the proof of Local Immersion Theorem in Guillemin & Pallock's Differential Topology on Page 15. But I got lost at the following statement: Define a map $G: U \times \mathbb{R}^{l-k} ...
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If $X$ is compact Hausdorff and $p\in X$, then there is a continuous $f:X\to\mathbb{R}$ that vanishes at $p$ and nowhere else

Prove or refute: If $p$ is a point in a compact Hausdorff space $X$ then there exists a continuous real-valued function $f:X\to\mathbb{R}$ that vanishes at $p$ and nowhere else.
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86 views

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 ...
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58 views

Researching for differential invariants

I have just graduated and I have to start thinking about topics for my PhD thesis and areas I am going to specialize in. The thing is that one thing that looks fun to me is classifying smooth ...
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92 views

Not well-defined parametrization of torus.

The problem statement: Exhibit explicit parameterizations covering $S^1 \times S^1 \subset \mathbb{R}^4$. My question: I had my attempt below, but my question lays at the very last. It seems ...
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Poincaré-Hopf theorem and its applications

I'm reading the basics of differential topology to try to understand the Poincaré-Hopf theorem, its proof and its applications. My plan is as follows: 1) Study transversality: its homotopy stability ...
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64 views

Does a smooth map from two disjoint spheres to one sphere exist?

In particular, are there two smooth maps $f : S^2 \cup S^2 \rightarrow S^2 \vee S^2$ and $g : S^2 \vee S^2 \rightarrow S^2$? What if there is an additional restriction that both spheres are the ...
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1answer
36 views

Check that $df_x(v) = (v,v).$

Here is a proof that I am totally different from my classmates'. So I am requesting for expert reference here. Thank you. :-) Let $f: X \rightarrow X \times X$ be the mapping $f(x) = (x,x).$ Check ...
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1answer
45 views

Does $\Gamma$ intersect $SL(2, \mathbb{R})$ transversely at $I$?

Identify the space of all $2 \times 2$ real matrices with $\mathbb{R}^4$ so that the matrix $\left( \begin{array}{cc} a & b\\ c & d\end{array} \right)$ corresponds to $(a, b, c, d)$. Let ...
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Why maximal atlas? [duplicate]

In the course of manifold, we use the maximal atlas to define the smooth or $ C^{ \infty } $ manifold. My question is: why maximal? (Of course, a maximal atlas is a $ C^{ \infty } $atlas ) When we ...
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1answer
128 views

Identification of each tangent space $T_pV$ with $V$ itself?

I found this statement from my text very confusing: What does it mean by identification of each tangent space $T_pV$ with $V$ itself? - what does "identification" really mean here? If it means ...
3
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1answer
65 views

How to find stable homotopy group given the quotient group?

If $\Theta_n$ is the group of exotic spheres in dimension $n$ and $\mathrm{bP}_{n+1}$ is the group of spheres that bounds parallelizable $(n+1)$-manifolds, $\pi_n^S$ is the $n$th stable homotopy group ...
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101 views

Groups of homotopy spheres II?

Where can I find "Groups of homotopy spheres: II", the sequel to "Groups of homotopy spheres: I"?
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195 views

Can we smoothly embed $\mathbb{S}^2 \times \mathbb{S}^1$ or $\mathbb{RP}^2 \times \mathbb{R}$ in $\mathbb{R}^4$?

I've been thinking a little bit about smooth embeddings recently. In particular, I was wondering: Do the $3$-manifolds $\mathbb{S}^2 \times \mathbb{S}^1$ and $\mathbb{RP}^2 \times \mathbb{R}$ ...