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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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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71 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 ...
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173 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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429 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
368 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$. ...
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1answer
144 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 ...
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1answer
59 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
978 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 ...
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1answer
184 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 ...
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89 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
45 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
68 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 ...
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1answer
89 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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566 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
164 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
68 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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235 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 ...
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167 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 ...
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1answer
116 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
41 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
169 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 ...
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1answer
73 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
96 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 ...
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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
195 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
40 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 ...
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217 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$X <$ ...
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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
96 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
74 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
67 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
98 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
69 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
223 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 ...
2
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1answer
534 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
209 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
185 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
110 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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0answers
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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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 ...
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1answer
100 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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292 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 ...
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1answer
183 views

Showing a diffeomorphism extends to the neighborhood of a submanifold

Does anyone have a proof of problem 14, on page 56 of Guillemin and Pollack? I meant to do it as an exercise (I'm teaching myself the subject) but I'm struggling with the last step. Suggestions? ...
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0answers
75 views

are closed orbits of Lie group action embedded?

Consider a smooth action $G\curvearrowright M$ of a Lie group on a manifold. Suppose that an orbit $G\cdot p$ is closed. Is the orbit an embedded submanifold. In general we know that the orbits are ...
2
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1answer
77 views

Uniqueness of “Punctured” Tubular Neighborhoods (?)

Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo) ...
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1answer
57 views

Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a real analytic function with infinitely many zeros. Let $a\neq 0$ be a real number. Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$.
4
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1answer
70 views

Smoothness in Banach space

I need a reference about a definition. Let $n$ be an integer and $G$ be a group of $H^n$(Sobolev) automorphisms of a vector bundle $E$ on some manifold $M$ and $C$ be the space of connections of class ...
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2answers
123 views

Application of the transversality theorem

I am trying to do this question in Bredon's Topology and geometry about using the transversality theorem to show that the intersection of two manifolds is a manifold. Now it goes as follows: Let ...