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 does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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17 views

Derivative of vector field on a manifold sends the tangent plane to itself

A vector field $\vec{v}$ on a (smooth) manifold $X \subset \mathbb{R}^N$ is a map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)\in T_x(X)$ for every $x \in X$. Suppose $\vec{v}(x)=0$, show that ...
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1answer
37 views

Let $f : M\to N$ be a submersion with $M$ compact and $N$ connected. The $f$ is surjective.

I have no idea how to do this. I tried think in, once $N$ is connected and locally path connected it has to be path connected, but, this does not help. Any hints, solutions, will be very appreciate... ...
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1answer
54 views

Example of a disconnected manifold where the tangent space is not the dimension of the manifold?

Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold. Well, is there an example of a simple disconnected manifold that doesn't ...
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30 views

De Rham cohomology ring of flag bundles/manifolds in Bott and Tu

I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working ...
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1answer
42 views

Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point $...
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2answers
71 views

Non-compact 3-manifold with incompressible boundary

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times [0,\...
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0answers
31 views

$f: M \to Y$ $C^{1}$ map between manifolds $M$ e $N$ with dimension $m$ and $n$ respectively then $f$ is locally proper

We say that $f:f: M \to N$ is locally proper if for all $x \in M$ there exist $V \ni x $ open in $M$ such that $f|_{\overline{V}}:\overline{V} \to Y$ is proper. I know two ways to prove it. First, ...
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1answer
25 views

Bundle that is isomorphic to the bundle of Whitney sum

I am involved with one question that a friend of mine asked. If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ ...
2
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1answer
56 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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1answer
82 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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1answer
27 views

Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point $...
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29 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
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22 views

Normal bundle of sub-manifold is a manifold

This is exercise 2.3.12 of Guillemin and Pollack: Let $Z$ be a sub-manifold of $Y$, where $Y \subset \mathbb{R}^M$. Define $N(Z;Y)=\{(z,v):z\in Z, v\in T_z(Y), v \perp T_z(Z)\}$. Prove that $N(Z;Y)$ ...
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1answer
31 views

How to show that is a submanifold or How to derivate the determinant function?

I am trying to show that the space of $2\times 2$ matrix with rank equals $1$ is a submanifold of $\mathbb{R}^4 - \{0\}$ whoose the dimension equals $3$. To do this, I have defined $\det : \mathbb{R}^...
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1answer
33 views

On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ $$D_A F(B) = \frac{\...
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2answers
77 views

Covering spaces as fiber bundles

I am reading Steenrod. I think there is something wrong or sloppy in his definition of a covering space as a fiber bundle. He writes: A fibre bundle consists of: (i) A topological space $B$ (ii) a ...
4
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1answer
37 views

The tangent space of the moduli space of connection?

I'm reading one of Floer's paper. (An Instanton-Invariant for 3-Manifold). Let $M$ be a $3$-manifold. A principal $SU_2$-bundle P over $M$ must be trivial. Fixed a trivialization $P \cong M \times ...
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0answers
17 views

covariant derivative of a helicoid

Given a helicoid $S$ parametrized by $x(u,v)=(v\cos(u),v\sin(u),u)$, a point $p=(1,0,0)$ on the helicoid, a tangent vector $v=(2,1,1)$ on $T_pS$ and a tangent vector field $X(u,v)=(v\cos(u),v\sin(u),0)...
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25 views

When the boundary of a manifold is orientable?

I am not sure whether the boundary of some manifold is definitely a manifold, but let's assume it is anyway. Then in what case the boundary is an orientable manifold. Maybe when the manifold can be ...
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26 views

Construct manifold from vector fields and point in $\mathbb{R}^n$

Suppose one is given a set of vector fields $X_{\alpha} \in \mathbb{R}^n$. Is it generally possible to construct a manifold embedded in $\mathbb{R}^n$ going through a particular point and whose ...
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41 views

Is there $f: U \to \mathbb{R}^{n}$ injective such that…

Let $f: U \to \mathbb{R}^{n}$ $C^{1}$ injective where $U$ is a open in $\mathbb{R}^{n}$ (so $f$ is open by invariance domain theorem). a) Is there exist $f$ such that dim $ker(df_{x}) >$ dim $...
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16 views

Show that if $M^n$ is a smooth manifold and $A,B$ are closed disjoint sets on $M$ then there is a smooth function $ 0\le f \le 1$

Show that if $M^n$ is smooth a manifold and $A,B$ are closed disjoint sets on $M$ then there is a smooth function $ 0\le f \le 1$ with $f(A) = 0$ and $f(B) = 1.$ What I am trying is: Note that $X-A$...
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2answers
84 views

Doubt in definition of symmetric continuous function and norm in Kupka's paper

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage: For $H=l^{2}$ "Let $H^{*}$ be the dual of $H$. A base ...
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1answer
75 views

Proof that this is a smooth manifold

Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^4$ be defined by $F(u,v) = (u+v, uv, u-v, v^3)$ and let $M = F(\mathbb{R}^2)$. Prove that $M$ is a smooth manifold. The proof that my TA posted online was ...
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2answers
41 views

Is an isometric and bijective mapping between two metric spaces complete?

If I have the two metric spaces $(X,d_x)$ and $(Y,d_y)$ with the mapping $f : X \to Y$ that is both an isometry and bijection between X and Y. How do I show that $(Y,d_y)$ is complete iff $(X,d_x)$ ...
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1answer
31 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
3
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2answers
67 views

Non-orientable manifolds and mod 2 homology

I was reading the wonderful book "The wild world of 4-manifolds" by Alexandru Scorpan and I found the following sentence: "We are able to orient $\mathfrak{M}$ (else we only get modulo 2 invariants)."...
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2answers
63 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
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0answers
24 views

Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
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How to calculate Fourier coefficient of $f\in C^{\infty} (\mathrm{T^3})$?

I was trying to calculate the $k$-th fourier coefficient $c_k$ of some smooth functions on $T^3$, say $k=(m,n,p)\in \mathbb{Z}^3$. In a write-up I found online, it has the following definition: $$c_k =...
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1answer
35 views

Visualizing sections of nontrivial vector bundles

My question is simply: how does one think about sections of nontrivial vector bundles on a smooth manifold, for example? The canonical example I think of is a vector field, i.e. a section of the ...
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2answers
63 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
34 views

Homotopy classes of manifolds of dimension $2n$ which are $(n-1)$ connected

In his essay "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres" Milnor states (in passing) "The manifold can be built up (up to homotopy type) by taking ...
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1answer
43 views

Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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0answers
24 views

Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
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1answer
79 views

Existence of incompressible surface in a non-orientable manifold.

Let $M$ be a compact $P^2$ -irreducible 3-manifold. If $M$ is non-orientable, then there is a compact surface $F$ properly embedded in $M$ such that $F$ is two-sided, non-separating and ...
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0answers
23 views

Chart-free definition of manifold

There is some way to avoid charts in the definition of a (topological, or smooth, or any other) manifold? The choice of a cover by charts are not really important for the manifold; many manifolds that ...
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1answer
48 views

Degree of map between surfaces of genus $g>1$ is $1$, $0$ or $-1$

Let $M$ be an orientable surface of genus $g>1$, I can assume compact. Let $f$ be a continuous map from $M$ to $M$. I want to prove that the degree of $f$ is $1$, $0$ or $-1$. For a surface of ...
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1answer
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I cannot understand an explanation why 2-sphere is simply connected.

I am studying Elementary Differential Geometry written by Barrett O'Neill. In page 188, Chapter 4.7, there is an explanation why 2-sphere is simply connected. The following is from the text : ======...
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73 views

Trouble with definition of signature of a compact manifold

The signature of a manifold, as I understand it, is defined as follows: Given a connected, compact, and oriented manifold $M$ of dimension $4n$, we may define a quadratic form on the cohomology group ...
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2answers
61 views

On the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$

Let $\exp$ be the exponential map on the Riemannian manifold M and $O$ is its domain in $TM$. Consider the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$, where $\pi$ is the ...
4
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1answer
54 views

If $M$ is a connected manifold, does $M\setminus\{p\}$ have finitely many components?

Let $M$ be a connected manifold and $p\in M$. Is it true that $M\setminus\{p\}$ has only finitely many connected components? (We can also suppose $M$ is compact if that helps.) I think this is ...
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1answer
22 views

Being morse function for a determinant map on M(n)

Show that the determinant map on M(n) is Morse function if n=2. I know that f to be a morse function, all critical points for f must be nondegenerate. But i dont know how i calculate the ...
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64 views

Smooth vs topological orientation

I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones: 1) Coherent pointwise orientation of the tangent spaces. 2) ...
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0answers
45 views

Using Poincare duality to show a closed manifold is a homology sphere

Suppose that $M$ is an orientable, compact, $(n-2)$-connected, $(2n-3)$-dimensional smooth manifold, where $n$ is a natural number. I want to show that $M$ is a homology sphere if and only if the ...
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1answer
33 views

Finding a basis for the cohomology vector space of 1-forms in the 2-torus, $H^1 (T^2)$

I would like help in understanding where I am going wrong here: If I consider the 2-torus $T^2 = S^1 \times S^1$ with an atlas $(\theta_1,\theta_2)$, I can define 2 closed 1-forms $\omega_1 = d\...
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0answers
48 views

Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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36 views

Do the level sets of a constant rank map give a foliation of the domain?

I'm working through Smooth Manifolds by Lee and came to problem 19-8, where you're asked to prove that the level sets of a submersion form a foliation of the domain. However when I try to solve this ...
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24 views

A new definition of a focal point

Let $X$ be a manifold lying in $\mathbb{R^n}$, with $dim(X) = n-1$. Define $h: N(X) \to \mathbb{R^n}$ by $h(x, v) = x + v$, where $$N(X) = \{(x, v) \in X \times \mathbb{R^n}: v \perp T_x(X)\}$$ is the ...