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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What happens in dimension 125?

In Differential topology 46 years later (page 807, bottom of left column) Milnor states that for $n \neq 4, 125, 126$ if the order of the stable homotopy groups $|\Pi_n|$ is known then we can compute ...
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53 views

Let $f:S^1\rightarrow \mathbb{R}^3$ be a smooth map with $df_x \neq 0$, for all $x \in S^1$.

Let $f:S^1\rightarrow \mathbb{R}^3$ be a smooth map with $df_x \neq 0$, for all $x \in S^1$. Show that there is a plane $C$ through the origin of $\mathbb{R}^3$ such that $p\circ f:S^1\rightarrow C$ ...
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146 views

What does the notation $\epsilon(f(x))s$ mean?

I am very, very confused with the notion $\epsilon(f(x))s$. To my understanding, $s$ is a map sends to $F(x,s)$, and $\epsilon$ is the distance function given a point $f(x)$. So what does ...
2
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181 views

Prove that a compact cone is not diffeomorphic to the 2-sphere

In Tapp's "Matrix Groups for Undergraduates" he briefly states (p.103) that a compact cone (he just shows a picture of a manifold with a ''cone point'') is not diffeomorphic to a 2-sphere. I would ...
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95 views

Finite set on compact manifolds

I feel blocked with this claim - it sounds intuitively true, just thinking as a jellyfish entering a real line, the intersection of her legs with the real line is certainly finite since the jellyfish ...
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35 views

Transverse a single point

I got very confused with understanding this theorem. So $\{y\}$ is a point, how could it be transversed by $f$? Proof: Given any $y \in Y.$ alter $f$ homotopically to make it transversal to ...
3
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1answer
219 views

Is [0,1] closed?

I thought it was closed, under the usual topology $\mathbb{R}$, since its compliment $(-\infty, 0) \cup (1,\infty)$ is open. However, then then intersection number would not agree mod 2, since it can ...
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76 views

F is a submersion of $X \times S$, and for a fixed $x$.

The following text confuses me very much: (1) For fixed $x \in X$, is my understanding correct? $F_{(x,s)}$ takes $(x,s)$ and gives $f(x)+s$, so $dF_{(x,s)}$ takes $(x,s)$ and gives $df(x)+ds$. But ...
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218 views

The proof in textbook on The Transversality Theorem

I was especially thrown out by the proof on The Transversality Theorem on the point "we want to exhibit a vector $v \in T_x(X)$ such that $df_s(v) - a \in T_z(Z).$" I understand so far that in order ...
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42 views

What is local classification theorems for immersions and submersions?

The term "local classification theorems for immersions and submersions" do not appear in the previous text, but may I assume they are just Local Immersion/Submersion Theorem (exists local ...
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224 views

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 ...
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66 views

If $f(x)$ is an extreme value, why $f$ cannot be a coordinate function?

In the text, it says: Consider smooth function on a manifold $f: X \to \mathbb{R}$. If $f(x)$ is an extreme value, then $f$ cannot be a coordinate function near $x$, so $df_x$ must be zero. I ...
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1answer
38 views

$f$ is either regular or $df_x = 0$.

In the text, it says: Consider smooth functions on a manifold $X$: $f: X \to \mathbb{R}$, at a particular $x \in X$, $f$ is either regular or $df_x = 0$. So I am not certain here: if $df_x \neq ...
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83 views

Help with Kervaire paper

I was trying to read Kervaire's 1960 paper where he first shows the existence of a manifold that does not admit differentiable structure and I got stuck. On the second page of the paper where he ...
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1answer
158 views

Exotic spheres and homotopy groups: sanity check

There is a homomorphism $\Theta_n \to \Pi_n/J_n$ where $\Theta_n$ is used to denote the group of diffeomorphism classes of $n$-spheres (with connected sum), $\Pi_n$ denote the $n$-th stable homotopy ...
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54 views

Motivation for Kervaire's seminal paper

Let DIFF denote the category of smooth manifolds, TOP the category of topological manifolds and PL the category of piecewise linear manifolds. In Kervaire 1960 it is shown for the first time that ...
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1answer
45 views

The map $S_n \to \Pi_n / J_n$

Let $S_n$ denote the set of all oriented diffeomorphism classes of closed smooth homotopy n-spheres. Let $S_n^{bp}\subseteq S_n$ denote the subgroup represented by homotopy spheres that bound ...
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201 views

Non-vanishing vector field on $\mathbb{R}P^{2n+1}$

I'm trying to cook up a non-vanishing vector field on $\mathbb{R}P^{2n+1}$. I know that $S^{2n+1}$ admits one, namely $(x_1,\dots,x_{2n+2})\mapsto (-x_2,x_1,\dots,-x_{2n+2},x_{2n+1})$. Moreover, I ...
3
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1answer
116 views

Bordism sanity check

Were framed cobordism and h-cobordism invented to use for different purposes? I have been slightly confused about all the different types of cobordism. Now I am wondering if h-cobordism was invented ...
3
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1answer
57 views

$J(X)$ 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 ...
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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
325 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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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 ...
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480 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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176 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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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
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1answer
111 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 ...
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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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433 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 ...
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2answers
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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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1answer
149 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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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 $\Omega(t_0, \dots, t_k)$ ...
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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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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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201 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
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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255 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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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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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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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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56 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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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 ...
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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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142 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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113 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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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, ...