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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Differentiable Version of the Jordan-Brouwer Separation Theorem

The Jordan-Brouwer separation theorem is a celebrated result of algebraic topology, which generalizes Jordan curve Theorem (it was proved independently by Lebesgue and Brouwer in 1911: see Dieudonné, ...
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76 views

Intrinsic definition of differential k-form on smooth manifold

Suppose I have a $k$-dimensional manifold embedded in $\mathbb{R}^n$. Munkres defines a $k$-form on $M$ as a a function $\omega$ that assigns an alternating tensor at each point $p \in M$ that acts on ...
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Sphere of normals

In my situation, I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. ...
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26 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
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42 views

Describe an atlas of smoothly related charts for the Special Orthogonal Group $SO(3)$

The Special Orthogonal Group $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, det(A) = 1}$} I have successfully shown that $SO(3)$ is a manifold, but I am having a difficult time explicitly finding a ...
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1answer
21 views

If a surface has a differentiable Gauss map, then it has an orientation?

If a surface $S\subset R^3$ has a differentiable Gauss map $N:S\rightarrow S^2$, then $S$ has an orientation? How can I prove this statement? (Here, orientation is defined by a choice of equivalence ...
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41 views

Question on the definition of outward normal vector from Spivak, Calculus on Manifolds

The following definition of the outward unit normal at the boundary of a manifold $M \subseteq \mathbb R^n$ is taken from Spivak, Calculus on manifolds (page 119). If $M$ is a $k$-dimensional ...
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36 views

The defintion of orientation of a manifold from Spivak, Calculus on Manifolds

In Spivak Calculus on Manifolds the author uses a definition of orientation of a manifold which I do not understand, and which I do not found elsewhere. I cite: It is often necessary to choose an ...
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64 views

Prove topological space has countable basis

Given a topological subspace M of $\mathbb{R}^2 \times S^1$ defined by $(x,y,e^{i\theta})$ and two charts $(U,h)$, $(V,k)$ such that $H:\mathbb{R} \times (-\pi,\pi) \to M$ $H(x,\theta) = ...
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1answer
27 views

A particular case of Sard's Theorem

I want to prove a particular case of Sard's Theorem, obviously without using the main Sard's Theorem. Let $f:M\to N$ be a differentiable ($C^{\infty}$) function. If $m=\dim M<\dim N=n$, then ...
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36 views

Question on how differential form as defined for subsets of $\mathbb R^n$ and integration on them in Spivak, Calculus on Manifolds

If $V$ is a vector space, denote by $\Lambda^k(V)$ the space of alternating multilinear maps from $V^k$ to $\mathbb R$, i.e. the space of alternating $k$-tensors. Also for a point $p \in \mathbb R^n$ ...
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100 views

If $\delta c = 0$, does it follow that $d\xi = 0$?

Let $\mathcal{U} = \{\mathcal{U}_\alpha\}_{\alpha = \infty}^\infty$ be a locally finite open covering of the manifold $M^n$, with smooth functions $\lambda_\alpha$, compactly supported in $U_\alpha$. ...
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1answer
66 views

De Rahm cohomology of a sphere, help with proof

I am working through Guillemin and Pollack's proof that the de Rahm cohomology of the sphere is $H^p(\mathbf{S}^k) = \mathbf{R}$ for $p = 0$ and $p = k$ and $H^p(\mathbf{S}^k) = 0$ otherwise. Here, ...
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47 views

If a manifold has a submanifold, then the local space is a cartesian product or splits in some other way?

The following definitions are taken from Marsden et al. Manifolds, Tensor Analysis, and Applications. Definition 1: Let $S$ be a set. A chart on $S$ is a bijection $\varphi$ from a subset $U$ of ...
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39 views

A question on an exercise to show that unit sphere could not be covered by a single chart

The following is an exercise from Marsden et al, Manifolds, Tensor Analysis, and Applications in the first chapter on manifolds. First let me cite three essential definitions: Definition 1: Let ...
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If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$

Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
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40 views

Fibration over contractible space is homotopic to a fiber

Let $\pi: E \to B$ be a fibration of $E$ over $B$, let $F = \pi^{-1}(b)$ for some $b \in B$ be a representative fiber, and suppose that $B$ is contractible. Is it always the case (or are there some ...
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48 views

Lie bracket of exact differential one-forms

Let $(M,g)$ be a Riemannian manifold. The musical isomorphisms $^\flat:\chi(M) \to \Omega^1(M)$ and $^\sharp:\Omega^1(M) \to \chi(M)$ allow the space of differential one-forms $\Omega^1(M)$ to be ...
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49 views

De Rham cohomology ring of the Stiefel manifold in low dimensions

Let $V_k(\textbf{R}^n)$ be the Stiefel manifold : the $k$-frames in the $n$ dimensional real space. I'm trying to understand the de Rham cohomology ring for $k=2$ or $k=n-1$. I had good ideas for ...
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40 views

Two vector bundles over same base manifold $X$

What are two vector bundles over the same base manifold $X$ which are isomorphic as vector bundles in the general sense, but not isomorphic over $X$? (That is to say, this would demonstrate that there ...
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68 views

Motivation of paracompactness

"A paracompact space is a topological space in which every open cover admits a locally finite open refinement" is the definition of paracompactness on Wikipedia. Comparing with the definition of ...
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31 views

Show that $\mathbb{PR}^1$ is diffeomorphic to $S^1$

I know how to construct the charts. I will have: $$\phi_i(x_1,x_2) : U_i \to \mathbb{R}$$ where $Y_i \subset \mathbb{RP}^1$ defined by: $U_i :=\{(x_1,x_2) : x_i > 0\},$ $\phi_1(x_1,x_2) = ...
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Can a gradient vector field with no equilibria point in every direction?

Suppose that $V:\mathbb{R}^n \to \mathbb{R}$ is a smooth function such that $\nabla V : \mathbb{R}^n \to \mathbb{R}^n$ has no equilibria (i.e. $\forall x \in \mathbb{R}^n : \nabla V (x) \not = 0$). ...
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Construct a diffeomorphism between $B_{\epsilon}(0)$ and $\mathbb{R}^n.$

Why does the map $\phi(y) := tg(\frac{\pi\|y\|}{2\epsilon})\frac{y}{\|y\|},$ if $ y \neq 0,$ and $0$ if $y = 0$ is a diffeomorphism between the open ball of radius $\epsilon$ and the entire ...
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125 views

Does there exist a smooth curve satisfying certain conditions?

Let $\gamma := dc - a\,db$ be a $1$-form on $\mathbb{R}^3$ with coordinates $a$, $b$, $c$. Have $\omega$ be the $2$-dimensional subbundle of $T\mathbb{R}^3$ consisting of vectors in the kernel of ...
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1answer
30 views

connection and change of local trivialization

$E\to B$ is a vector bundle. $\nabla$ is a connection of this bundle. Choose a local trivialization of $E$, then we can write a formula of $\nabla$ with respect to this local trivialization: ...
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51 views

When does a codimension 1 submanifold admit a transverse vector field?

I'm having some trouble with the following problem, which comes from a released qualifying exam: Assume that $N \subset M$ is a codimension 1 properly embedded submanifold. Show that $N$ can be ...
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33 views

Restrictions of $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if X is paracompact

The following assertion appears in Milnor's Characteristic Classes. The restrictions of a vector bundle $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if $X$ is paracompact. ...
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Exist vector field having only finitely many zeros, all lying in open set of compact connected manifold?

Let $U$ be any open set on the compact connected manifold $X$. Does there exist a vector field having only finitely many zeros, all of which lie in $U$?
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209 views

A map of a compact connected manifold with non-empty boundary to a sphere of the same dimension is null-homotopic

In Milnor & Kervaire's Groups of Homotopy Spheres paper, this claim: A map of a compact connected manifold with non-empty boundary to a sphere of the same dimension is null-homotopic is made ...
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1answer
51 views

Unique way to show $S^n$, $n \geq 2$ is simply connected.

This questions is asked in Armstrong's Topology book, and I am totally stuck.... I could really use a major hint: Think of $S^n \subset \mathbb{E}^{n+1}$. Given a loop $\alpha \in \pi_1 (S^n , ...
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28 views

Is range of completely continuous of bounded set finite dimensional set?

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Let $\Omega$ is a bounded set of ...
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Extension theorem from Guillemin-Pollack, motivated sketch of proof?

Let $W$ be a compact, connected, oriented $k + 1$ dimensional manifold with boundary, and let $f: \partial W \to S^k$ be a smooth map. Could anybody sketch with good motivation that $f$ extends to a ...
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1answer
40 views

The circle does not intersect the unbounded component of $\mathbb{R}^2 \setminus C$

We get the following problem in our differential geometry class. Let $ C $ be a smooth, non-degenerate simple closed curve traveling counterclockwise. Suppose that the curvature $ \kappa $ of C is ...
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28 views

Topology of complex variety $Y^2=F(X)$

Suppose $F\in\mathbb C[X]$ is a polynomial of degree $n=2k$ where $k\ge2$ is an integer. Suppose further that the $n$ roots $\alpha_1,\ldots,\alpha_n$ are distinct. Then, consider the algebraic subset ...
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55 views

Does the set of diffeomorphisms which are induced by flows form a group?

Let $M$ be a smooth manifold. Consider the set of diffeomorphisms which are induced by flows of vector fields. (which are not time-dependent) Is this set a subgroup of $\text{Diff}(M)$? (Note that ...
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Rotation of a planar vector field

This is a square grid of 2D vectors. How do I compute the rotation number of each cell (composed by four 2D vectors) to discover critical points? I need to write a C++ function that computes it, ...
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31 views

Flow of Normalized Gradient Field of a Smooth Function

Suppose $f: \mathbb R^n \rightarrow \mathbb R$ is a smooth function, with a finite number of critical points (which are then isolated). Let us take as a vector field on $\mathbb R^n$ (minus those ...
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Subvariety in $\mathbb R^n$

I'm reading a lecture note (in French). They define the following: Let $S$ be a subset of $\mathbb R^n$ and $d \in \mathbb N$. $S$ is a sub-variety of class $\mathcal C^1$ and of dimension $d$ if for ...
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23 views

Confusion on Homogeneity Lemma

I am reading the Topology from the Differential View written by Milnor, and stuck at the Homogeneity Lemma. For the special case $N=\mathbb S^n$ the proof is easy: simply choose $h$ to be the ...
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50 views

Topological manifold but not “smooth”

I would like to know if there are examples of topological manifolds that not admit a smooth atlas, or more generally a differentiable structure of class $C^k$ for $k\geq1$. In the book "Introduction ...
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Is $H([f]) = \int_{S^{2n - 1}} \alpha \wedge d\alpha$ independent of all choices, defines a map $H: \pi_{2n - 1}(S^n) \to \mathbb{Z}$?

Let $[f] \in \pi_{2n - 1}(S^n)$. Choose a smooth representative $f: S^{2n - 1} \to S^n$. Let $\omega$ be a smooth $n$-form on $S^n$ with$$\int_{S^n} \omega = 1.$$We know that$$f^*\omega = d\alpha$$for ...
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Smooth representative $f: S^{2n - 1} \to S^n$, do we have $f^*\omega = d\alpha$?

Let $[f] \in \pi_{2n - 1}(S^n)$. Choose a smooth representative $f: S^{2n - 1} \to S^n$. Let $\omega$ be a smooth $n$-form on $S^n$ with$$\int_{S^n} \omega = 1.$$Do we have that$$f^*\omega = ...
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21 views

A disk cross sphere in a sphere

Let $\lambda$ and $\mu$ be positive integers and define $D(\lambda, \mu) = \{ p \in S^{\lambda + \mu -1} : \sum_{i \leq \lambda} x_i^2 \geq \sum_{i > \lambda} x_i^2 \}$. Why is $D(\lambda, \mu)$ a ...
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Two neighborhoods of $0$ in the plane and the upper half plane (resp.) cannot be diffeomorphic. [duplicate]

Let $\mathbb{R^2}$ be equipped with the standard topology, and let $\mathbb{H^2}$ be the upper half plane (containing the x-axis), equipped with the subspace topology. Let $U$ be an open neighborhood ...
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105 views

Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?

Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
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43 views

Two smoothly homotopic smooth maps induce same maps on de Rham cohomology

Let $a$, $b: M \to N$ be smoothly homotopic smooth maps. How do I see directly that $a$ and $b$ induce the same maps on de Rham cohomology? I know I want to construct a suitable chain homotopy between ...
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30 views

2-connected 6 manifolds with boundary $S^5$

What are the 2-connected 6-manifolds that have boundary $S^5$? Are they all of the form $(\sharp_{i=1}^k S^3 \times S^3) \backslash D^6$ for some $k \ge 1$? Also, I think if $M^5$ is ...
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1answer
62 views

Wedge product of closed form each with integral periods has integral period?

Suppose $\alpha$ and $\beta$ are closed forms on $M$ which have integral periods, i.e. for all $[A] \in H_*(M, \mathbb{Z})$ represented by a smooth cycle $A$, we have $\int_A \alpha \in \mathbb{Z}$, ...
3
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59 views

how to prove that every low-dimensional topological manifold has a unique smooth structure up to diffeomorphism?

It is a deep fact of low-dimensional manifold topology that the notion of isomorphism coincides for the three categories Top, PL, and Diff. In other words, every topological manifold of dimension ...