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

(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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72 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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886 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
860 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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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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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 ...
3
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1answer
205 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
209 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
34 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
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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
91 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
103 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
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1answer
60 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 ...
1
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1answer
114 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 ...
6
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782 views

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 ...
0
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1answer
40 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 ...
2
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1answer
48 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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0answers
82 views

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 ...
3
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2answers
168 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
69 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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2answers
124 views

Groups of homotopy spheres II?

Where can I find "Groups of homotopy spheres: II", the sequel to "Groups of homotopy spheres: I"?
8
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231 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 $\mathbb{RP}^2 \times \mathbb{R}$ ...
1
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1answer
170 views

GP 1.2.10(b) The tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$

If $\triangle$ is the diagonal of $X \times X$, show that its tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ I don't have the slightest idea on how to do this. ...
1
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1answer
187 views

Which compact (orientable) surfaces are parallelizable?

Which compact (necessarily orientable) smooth $2$-manifolds are parallelizable? I'm aware that the sphere $\mathbb{S}^2$ is not parallelizable, whereas the torus $\mathbb{T}^2 = \mathbb{S}^1 \times ...
2
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2answers
182 views

GP 1.2.2 $T_x(U) = T_x(X) \text{ for } x \in U.$

This is exercise 1.2.2 on Guillemin and Pollack's Differential Topology If $U$ is an open subset of the manifold $X$, check that $$T_x(U) = T_x(X) \text{ for } x \in U.$$ I am fairly confused ...
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401 views

Showing Degree $1$ Maps Induce Surjections on $\pi_1$

I am running a qualifying exam prep course. A question I posed to my students was: Suppose that $M$ and $N$ are compact, oriented manifolds and $f:M\longrightarrow N$ is of degree $1$. Show that ...
12
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0answers
332 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
4
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1answer
742 views

Immersion is a diffeomorphism

Suppose $X$ is a smooth, compact, connected $n$-manifold without boundary which admits an immersion to $S^n$. Show that if $n>1$, then this immersion is a diffeomorphism. Thanks for the very ...
2
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2answers
122 views

The matrix specify an algernating $k$-tensor on $V$, and dim$\bigwedge^k(V^*)=1$

The end of the hint "The matrix specify an algernating $k$-tensor on $V$, and dim$\bigwedge^k(V^*)=1$" does not make sense to me. In my not very assured understanding, the $k$-tensor ...
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26 views

The anticommutative multilinear form interchanges with neighbors right?

On page 155 of Guillemin and Pollack's Differential Topology, it says: A tensor $T$ is alternating if the sign of $T$ is reversed whenever two variables are transposed: $$T(v_1, \ldots, v_i, \ldots, ...
1
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1answer
101 views

Rôle of smooth structure

A central problem of interest in topology is the calculation of $\pi_n(S^m)$ and the classification of manifolds in general. In 1961 Kervaire constructed a manifold that does not admit a smooth ...
0
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1answer
461 views

wedge product and determinant

I don't really know what $[\phi_i(v_j)]$ really is. As far as I understand, $\phi_i$ is a linear transformation - a matrix; and $v_j$ is the column vector it eats. So $[\phi_i(v_j)]$ spit out column ...
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2answers
57 views

All 1-tensors are alternating

This statement from page 155 of Guillemin and Pollack's Differential Topology. I would assume because 1-tensors can not alternate because they have nothing to alternate with, so they are ...
5
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0answers
60 views

Which is harder to compute: $\pi_{n+k}$ or $\Omega^{fr}_n$?

Denote the $n+k$-th homotopy group of $S^n$ by $\pi_{n+k}(S^n)$ and the group of framed cobordism classes by $\Omega_n^{fr}(S^k)$. A central problem of algebraic topology is to compute $\pi_i(S^j)$ ...
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63 views

How is “index” at an Walrasian equilibrium proved? (in relation to Hopf-Poincare theorem)

So, the index of an (Walrasian/general equilibrium) equilibrium point is determined as the sign of $(-1)^{L-1} \times \det M$ where $M$ is a matrix and $M_{ij} = \frac{\partial{Z_i}}{\partial {p_j}}$, ...
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2answers
89 views

Regarding orientation and orientation-reversing in local diffeomorphism

I am confused about orientation and orientation reversing in local diffeomorphism $f$ from manifold $X$ to $Y$ at some points. So, what does $f$ orientation-reversing at a point mean?
5
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1answer
176 views

Relation between quadratic refinement and quadratic form

The question in the title has now been bothering me for days. I first came across the term quadratic refinement when I read about the Kervaire invariant when reading Kervaire's 1960 paper. The ...
3
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1answer
271 views

Importance of triangulation

Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed." What is the ...
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1answer
127 views

How does one prove that local diffeomorphism is submersion?

How does one prove that local diffeomorphism is submersion? For a manifold, what does it being disconnected mean? I get what "disconnected" means for a graph, but not for a manifold.
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2answers
93 views

Does $\mathrm{Mat}_{m \times n}$ have boundary?

To me, $\mathrm{Mat}_{m \times n}$ is isomorphic to $\mathbb{R}^{mn}$, hence is boundaryless. But this disqualified the use of Sard's theorem in this question: An exercise on Regular Value Theorem. ...
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1answer
81 views

Why $\Sigma$ is minimal, if $\frac{d}{dt} |_{t=0} \mathrm{Area}(\Sigma_t)=0$?

In this work http://arxiv.org/pdf/1204.2883v1.pdf Martin Li claimed that $\Sigma\subset M$ is minimal and $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$ if, only if, $$0 = ...
6
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1answer
499 views

A Surjective Local Smooth Diffeomorphism That is Not A Covering Map

Let $\pi:M_1\rightarrow M_2$ be a surjective $C^{\infty}$ map between two connected manifolds with $d\pi$ an isomorphism. If $M_1$ is compact, it is seen that $|\pi^{-1}(m_2)|$ is finite, so $\pi$ is ...
0
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1answer
45 views

Smooth immersion $\gamma$ from $\mathbb R \to \mathbb R^2$ such that $\operatorname{Im}\gamma=\{(x,\lvert x\rvert)\mid x\in \mathbb R\}$

Can I find a smooth immersion $\gamma$ from $\mathbb R \to \mathbb R^2$ such that $\operatorname{Im}\gamma=\{(x,\lvert x\rvert)\mid x\in \mathbb R\}$, I cannot take $\gamma(t)=(t,\lvert t\rvert)$ as ...
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1answer
405 views

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 $\epsilon > 0$ ...
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134 views

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 ...
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1answer
194 views

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 $f + Ax$ ...
3
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1answer
80 views

Showing that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is a smooth vector bundle over $S^1$

I want to show that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is the total space of a smooth vector bundle over $S^1$. Here $\sim$ is defined as follows: fix a linear isomorphism $L$ of ...
5
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2answers
557 views

Show that $f(C)$ has Hausdorff dimension at most zero.

We say that a set $A \subset \mathbb{R}^n$ has $d$-dimensional Hausdorff measure zero if for all $\epsilon > 0$ there exists a covering of $A$ by countably many cubes $S_i$ with side lengths $s_i$ ...
0
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1answer
24 views

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 ...
13
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698 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...