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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$J(X)$ and exotic spheres.

I read that we can realize exotic sphere as the coker of $J$-homomorphism. SO we can consider the exotic sphere $S^7$ realized using an identificantion of $B^4 \times S^3$ (where I donote the four ...
-2
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197 views

Difficulty in Sard's theorem [closed]

In Lee's Int to smooth manifolds, p 130 (Step 1 in the proof of Sard's theorem) line 18-19 "Because ..." is not clear to me. I shall be very much thankful, *strong text*if some one make it clear to ...
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2answers
331 views

The tangent space to the preimage of $Z$ is the preimage of the tangent space of $Z$.

More generally, let $f: X \to Y$ be a map transversal to a submanifold $Z$ in $Y$. Then $W = f^{-1}(Z)$ is a submanifold of $X$. Prove that $T_x(W)$ is the preimage of $T_{f(x)}(Z)$ under the ...
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2answers
62 views

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 ...
3
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1answer
486 views

The tangent space to the intersection is the intersection of the tangent spaces.

Let $X$ and $Z$ be transversal submanifolds of $Y$. Prove that if $y \in X \cap Z$, then $$T_y(X \cap Z) = T_y(X) \cap T_y(Z).$$ ("The tangent space to the intersection is the intersection of ...
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177 views

What does is mean by differentiate a matrix $E - DB^{-1}C$?

The problem I am trying to solve is: Prove that the set of $m \times n$ is matrices of rank $r$ is a submanifold of $\mathbb{R}^{mn}$ of of codimension $(m - r)(n -r)$. [HINT: Suppose, for ...
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0answers
45 views

derivative of matrix [duplicate]

Is the nonsingular matrices open? How can I show that every $m \times n$ matrix is in the image of the derivative of an $m \times n$ matrix (how to differentiate it?) Thanks
1
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1answer
112 views

Fundamental theorem of Morse theory for $\Omega(S^n )$

Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can ...
4
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2answers
150 views

Minimal geodesics in $S^{n+1}$

Let $\Omega^d(M)$ the space of minimal geodesics on a smooth manifold $M$. How can I prove that if $M= S^{n+1}$, $\Omega^d(S^{n+1}) \simeq S^n$?
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2answers
448 views

The tangent plane of orthogonal group at identity.

Why the tangent plane of orthogonal group at identity is the kernel of $dF_I$, the derivative of $F$ at identity, where $F(A) = AA^T$? Thank you ~
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89 views

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 ...
2
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2answers
85 views

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 ...
1
vote
1answer
153 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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0answers
29 views

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 $\Omega(t_0, \dots, t_k)$ ...
1
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1answer
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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78 views

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 $g_1, \dots, g_l$, ...
1
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1answer
203 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 ...
2
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1answer
93 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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1answer
261 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 ...
0
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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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125 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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115 views

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 ...
1
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1answer
103 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 ...
2
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0answers
57 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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0answers
35 views

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 ...
2
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0answers
158 views

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 ...
7
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2answers
143 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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115 views

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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1answer
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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1answer
68 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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0answers
104 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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22 views

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 $G: \mathbb{R}^2 \rightarrow S^1 \times S^1, G = g \times g$ ...
2
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1answer
116 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 ...
0
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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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40 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 ...
2
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1answer
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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0answers
63 views

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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638 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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3answers
665 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 ...
3
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0answers
60 views

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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0answers
46 views

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 ...
2
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1answer
156 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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2answers
197 views

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: ...
0
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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 ...
2
votes
1answer
51 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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3answers
88 views

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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1answer
97 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 ...
3
votes
1answer
59 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 ...