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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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
77 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 = ...
4
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1answer
326 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 ...
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1answer
44 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
362 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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0answers
116 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
171 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
75 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 ...
4
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2answers
344 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 $s_i$ ...
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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 ...
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0answers
518 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 ...
3
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1answer
466 views

Prove that a standard torus is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$

I was asked to prove that a standard torus(which means we don't consider those pathological cases where it intersects with itself, e.g horn torus) is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$. ...
4
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1answer
166 views

Show that the tangent space of the diagonal is the diagonal of the product of tangent space

I'm stuck on this question for quite a few days and still haven't got a clue what to do. The question is as follows: If $\Delta$ is the diagonal of $X\times X$ where $X$ is a manifold, show that its ...
0
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1answer
62 views

Intuitive idea of tangent space and definition coincide?

Let $M$ be a submanifold of $\mathbb{R}^n$ of codimension 1. Suppose you take $V\le \mathbb{R}^n$ a vector space of dimension $n-1$ and let $w \in \mathbb{R}^n\setminus\{0\}$ be an orthogonal vector ...
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2answers
1k views

Tangent Space of Product Manifold

I was trying to prove the following statement(#9(a) in Guillemin & Pollack 1.2) but I couldn't make much progress. "Show that for any manifolds $X$ and $Y$, $$T_{(x,y)}(X\times Y)=T_x(X)\times ...
6
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1answer
192 views

Is a continuous map between smoothable manifolds always smoothable?

Let $X$ and $Y$ be topological manifolds and $f:X\to Y$ a continuous map. Suppose $X$ and $Y$ admit a differentiable structure (at least one). My question: is it always possible to choose a ...
4
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2answers
91 views

Find a manifold which contains embedding of $K_5$

$K_5$ graph is not planar . I was asked to find a manifold which contains embedding of $K_5$ and use $5$ squares to represent $K_5$ "on" my new manifold. Embedding means that it can be drawn on the ...
3
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0answers
46 views

Differentiability of $\operatorname{dist}(x,\partial \Omega)$ function [duplicate]

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary and set $$\phi(x)=\operatorname{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|$$ for $x\in ...
3
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1answer
70 views

Axiomatizing oriented cobordism

According to the nLab entry for abstract cobordism categories, the natural way of axiomatizing the relation of two oriented manifolds being cobordant is the following: Definition 1 Two objects ...
2
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1answer
97 views

Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
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2answers
651 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
4
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1answer
186 views

Bounded vector field has globally defined flow

Let $X$ be a vector field on $\mathbb R^n$, and suppose that $\|X\|$ is bounded, where the norm is taken with respect to the Euclidean inner product. I am trying to show that $X$ has globally defined ...
3
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1answer
70 views

A proof on smooth function that I don't know what to proof.

Here's the question: Suppose $f: U \rightarrow V$ is a smooth map, for $U \subset R^k$ and $V \subset R^\ell$ open sets. That is, all partial derivatives (of all orders) of $f$ exist and are ...
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1answer
263 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
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1answer
39 views

Prove closed of dimension one of $X\times I$.

Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow R$ is Morse for every $t$. Show that the set $C = \{(x,t)\in X\times I : d(f_t)_x = 0\}$ forms ...
0
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1answer
168 views

Families of Morse functions

I don't have a clue with this problem. Thank you very much for your help & guidance. (a) Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow ...
1
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1answer
126 views

Fubini Theorem for measure zero

I know Fubini Theorem in calculus, but the measure zero version does not make sense to me: $n=k+1$, and $V_c$ is the "vertical slice" {c}$\times R_l$. Let $A$ be a closed subset of $R^n$ such that $A ...
2
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1answer
42 views

Show the points $u,v,w$ are not collinear

Consider triples of points $u,v,w \in R^2$, which we may consider as single points $(u,v,w) \in R^6$. Show that for almost every $(u,v,w) \in R^6$, the points $u,v,w$ are not collinear. I think I ...
3
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1answer
180 views

Thom class: Why are the two definitions equivalent?

We know that the Thom class $\tau_W$ is defined on a disk bundle $W\rightarrow L$, where $L$ is a $p$-dimensional manifold and the rank of $W$ is $k$. Let $[W]_0$ denote the fundamental class of the ...
0
votes
1answer
81 views

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 ...
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1answer
106 views

Eigenvalue of f and df

Given 1 is not an eigenvalue of $df$ at $x_0$, take a chart $(U,\phi)$ around $x_0.$ Then in this coordinate neighborhood, think of $f$ as a map from open ball in $\mathbb{R}^n$ (say $B$), to itself ...
2
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0answers
73 views

Homomorphisms of Lie Groups

$SO(3)$ denotes the special orthogonal group, which is the (open) subset of $O(3)$ on which the determinant is one. I have shown that every element of $SO(3)$ fixes a line in $\mathbb{R}$ pointwise ...
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1answer
245 views

Finitely many Lefschetz fixed points

The questions is Show that if $X$ is compact and all fixed points of $X$ are Lefschetz, then $f$ has only finitely many fixed points. n.b. Let $f: X \rightarrow X$. We say $x$ is a fixed point of ...
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2answers
41 views

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 ...
5
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2answers
255 views

Injectivity of a map between manifolds

I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ ...
0
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1answer
34 views

If $f$ is a Morse function, then so is $f \circ \phi^{-1}$, where $\phi: U \rightarrow \mathbb{R}^k$ is the coordinate chart.

I am trying to show: if when $f^\prime = 0$, then $f^{\prime\prime} \neq 0 \Leftrightarrow (f \circ \phi^{-1})^\prime = 0$, $(f \circ \phi^{-1})^{\prime\prime} \neq 0$. But the problem is, because ...
3
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1answer
103 views

Morse Function Definition: Does it implies Morse function is $C^2$?

In my understanding, Morse function just means the determinant of Hessian matrix is nonsingular at critical points. So my claims are: the function itself should be continuous the reference to ...
0
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1answer
79 views

Prove that the tangent space of a hyperplane is itself

I know this might sound really stupid: I was trying to show that the tangent space of a hyperplane is itself. I started by parametrising the hyperplane locally at $x$ with a diffeomorphism $\phi : U ...
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2answers
76 views

Definition of diffeomorphism functions

I see the definition of Diffeomorphism in Wikipedia homepage, but I don't understand whether "the differential of f (Dfx : Rn → Rn) should be bijective at each point x in U" or "f itself" should be ...
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1answer
104 views

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. ...
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0answers
76 views

Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
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1answer
265 views

uniformization theorem - squares and circles

I am trying to understand the uniformization theorem and get some intuition about it. The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...
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1answer
631 views

Prove that the tangent space has the same dimension as the manifold

I asked this question a couple of days ago. And I thought that I totally understood the question. However it turned out that I didn't, since the argument I constructed was proved to be wrong just now: ...
8
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1answer
254 views

Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
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1answer
229 views

Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k

In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
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1answer
113 views

How many Borel conjectures are there

The following may be referred to as Borel conjecture: Every strong measure zero set of reals is countable. On the other hand Wikipedia refers to the following as the Borel conjecture: Let $M$ and ...
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61 views

Kernel of the differential and weak topology

Suppose you have a differentiable map $\Phi : E \rightarrow F$ where $E$ and $F$ are Banach spaces, and a curve $t \mapsto u(t)$ of elements of $E$, with $u(0)=0$ and $\Phi(u(t))$ constant, such that ...
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0answers
86 views

On the density of vector fields with only nondegenerate zeroes

Suppose we have a manifold $X$ embedded in $\mathbb{R}^n$. Define the vector field $v_u(p) : X \rightarrow TX$ by taking the point $u \in \mathbb{R}^n$ to its natural (orthogonal) projection onto ...
3
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1answer
110 views

question from hatcher basic 3 manifolds

The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M? I had this problem reading ...
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0answers
362 views

Soft question: why are there non-smooth manifolds?

Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...