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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Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
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Smooth maps homotopic to the inclusion - updated.

First problem - (my original question before the editing) Prove or disprove the following: Let $A$, $B$ be differentiable manifolds such that $A \subseteq B$, and $s: A \to B$ a smooth map. Then $s ...
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Smooth homotopy

Let $M,N$ be manifolds. Suppose that $f_0, f_1:M\stackrel{C^\infty}\to N$ are homotopic, i.e. there exists a continuous mapping $f:M\times[0,1]\to N$ s.t. $f(x,0)=f_0(x)$, $f(x,1)=f_1(x)$. Then is ...
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Embedding counterexample

Lee writes on page 156 of Introduction to Smooth Manifolds: A smooth embedding is a map that is both a topological embedding and an immersion, not just a topological embedding that happens to be ...
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153 views

How to orient a manifold in the Euclidean space?

I learned that the orientation of a smooth manifold is a smooth choice of an orientation for bases of tangent space. Also I sometimes read that an embedded manifold in $\mathbb{R^3}$ inherites an ...
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Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$

Prove that a map $f:S^1→S^1$ extends to the whole ball $B=\{|z|≤1\}$ if and only if $deg(f)=0$ again I can only do one direction => $f:S^1→S^1$ is smooth, and $S^1 = \partial B$. Assume that ...
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Why is the pole generally outside the contour loop when its outside the contour loop in 2D?

The following contour integral is path dependent with the following results \begin{align} \oint_C\dfrac{dz}{z} = \begin{Bmatrix} 2\pi i && \text{when $z=0$ is inside C} \\ 0 && ...
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88 views

Prove that the Möbius band is not orientable.

Prove that the Möbius band is not orientable. I know that in the Möbius band the central circle is orientable. If I let $Y$ be the Möbius band and $Z$ be a compact submanifold of $Y$ with ...
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53 views

Suppose that $Z$ is a hyperspace in the oriented manifold $Y$, as sub manifold of codimension $1$. Prove that following statement are equivalent

Suppose that $Z$ is a hyperspace in the oriented manifold $Y$, as sub manifold of codimension $1$. Prove that following statement are equivalent a) $Z$ is orientable b) There exists a smooth field ...
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254 views

Coordinate-free definition of integration of differential forms?

Let $\omega$ be an $n$-form on an oriented $n$-manifold $M$. To integrate $\omega$, we choose an atlas $(O_\alpha, (x^1_\alpha,\dots, x^n_\alpha))_\alpha$ for $M$ and a partition of unity ...
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151 views

Extension of a map $g:\overline{B_1^n}\to \mathbb{R}^2$

Let $B_r^n\subset \mathbb{R}^n$ ($n\geq 6$) be the open ball with radius $r$ and let $g:B_2^n\to \mathbb{R}^2$ be an analytic map. How to define a continuous map $h:\mathbb{R}^n\to \mathbb{R}^2$ such ...
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82 views

Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory. It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that ...
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43 views

Smooth structure on $M\cup_f N$?

Let $M$ and $N$ be two smooth manifolds with $$\textrm{dim}(M)=\textrm{dim}(N)=n.$$ Let $U\subseteq M$ and $V\subseteq M$ be two open sets and $f:U\longrightarrow V$ a smooth diffeomorphism. Consider ...
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50 views

Prove that the index of a vector field is well-defined.

Consider first an open set $U \subset \Bbb{R^m}$ and a smooth vector field $v : U\to\Bbb{R^m}$ with an isolated zero at the point $z \in U$. The function $\overline{v}(x) = v(x)/\|v(x)\|$ maps a small ...
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210 views

For which k,n the k-covector is decomposable (14-2 from Lee)

This is homework so no answers please The problem is: Find for which k, n, a k-alternating map $\omega$ can be written as $\omega=\omega_{1}\wedge...\wedge \omega_{k}$ were $\omega_{i}$ are ...
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111 views

Submersion surjective on the complex projective space $\mathbb{C}P^1$.

If $S^3=\{ (z_1,z_2)\in\mathbb{C}^2\mid \vert z_1\vert^2+\vert z_2\vert^2=1\}$ and $\pi:S³\rightarrow\mathbb{C}P^1$ for $(z_1,z_2)\mapsto [(z_1,z_2)]$ since $[(z_1,z_2)]=\{ ...
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Definition of the hessian as a bilinear functional on the tangent space

In Milnor's Morse Theory, the Hessian of a smooth function $f : M \to \mathbb R$ defined on a manifold $M$ at a critical point $p$ is the bilinear functional on $T_p M$ defined as follows: ...
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Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, ...
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Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
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51 views

Orientability of integrable plane fields

Let $\xi \subset \text{T}M$ be a integrable plane field on a smooth 3-manifold (i.e. the tangent field of a foliation). Is it true that $\mathcal{F}$ is coorientable?
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Realizing a Contact Structure on S^1 x S^2 via an Open Book Decomposition

I am trying to learn about Contact Geometry and Open Book Decompositions. I went through the example of the Hopf Fibration for $S^3$ and how you can see a contact structure. I am now trying to do ...
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52 views

Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$( more detail)

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a ...
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59 views

Something about Degree of Map

I have been told that degree of map can be defined by the combinatorial method instead of the differential one. Assume $M$, $N$ is orientiable pseduomanifold, and $M$ is compact, $N$ is connected. Let ...
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Is a diffeomorphism's image automatically open?

Sorry if this question is trivial, I am new to smooth manifold theory. Let $\varphi : I \times \mathcal S^{n-1} \to X$ be a diffeomorphism. $I=(0,1)$, $\mathcal S^{n-1}$ is the unit sphere in ...
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106 views

Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

For $X^k\subset M^n$ compact submanifold with $\perp X$ trivial and set $S^k$ the $k$-sphere. Then there is a function $f:M^n\rightarrow S^k$ such that $X$ is the preimage for a regular value. My ...
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1answer
51 views

A function from a smooth manifold with boundary to $[0,\infty)$

Suppose $M$ is a smooth manifold with boundary, show that there exists a smooth function $f: M \rightarrow [0, \infty)$ such that $\partial M = f^{-1}(0)$. My attempt is that given a chart ...
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198 views

Connected sum of orientable manifolds

I was reading through Lee's Smooth Manifolds on the part regarding orientations and I was wondering if the connected sum preserves the orientability of manifolds. Intuitively it seems to be true, but ...
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41 views

Arbitrary Smooth structure

Is it possible to give a smooth structure to any objects? Say two lines intersecting at a point. It seems there is a smooth structure though at the intersecting point it is not locally euclidean if ...
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1answer
38 views

holomorphic function and simple zeros

How can I prove this? If $f$ is a holomorphic function in a domain $U$ and $f'(z)\neq0$ for all $z\in U$ then every zero of $f$ are simple and positive. Definition: $q\in U$ is a simple ...
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101 views

Does the compact manifold $f=0$ resist small perturbations?

Suppose we have a compact manifold of the form $\left\{f=0\right\}$ where $f:\mathbb R^n\to\mathbb R$ is a smooth Morse function. I am interested in showing that the manifold is topologically ...
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79 views

The index of a smooth vector field is well-defined

How can I prove that the index of a smooth vector field is well-defined? All I know that it is locally constant. Definition. Given an open set $U\subset\Bbb{R^m}$, and a smooth vector field ...
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1answer
249 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I've proved, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let $\gamma:[0,1]\rightarrow ...
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179 views

Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation.

Without using the Borsuk-Ulam theorem. Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation. I know that $f$ map anitpodal ...
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Question on the proof of the Poincare-Hopf theorem

OP is reading Milnor's celebrated Topology from the Differentiable Viewpoint's 6th chapter, where he deals with indices of vector fields and in particular the Poincare-Hopf theorem. A lemma he used is ...
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179 views

Poincare Hopf Theorem

I'm trying to apply the Poincare-Hopf theorem for a vector field over a closed disk. The vector fields sometimes have zeros on the boundary (if number of zeros is infinite, then it's zero over the ...
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1answer
85 views

Piecewise differentiable homotopy

Let $\gamma_0$ and $\gamma_1$ be piecewise differentiable closed curves in an open set $U$ in the plane $\mathbb{C}$. (Of course, you may consider a more general setting if relevant.) Suppose that ...
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Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent

Prove that the Borsuk-Ulam theorem and following ttheorem are equivalent Borsuk-Ulam theorem: Let $f:S^k \to R^{n+1}$ be a smooth map whos image does not contain the origin, and supposed that $f$ ...
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Is $M$ is a compact manifold?

Let $M$ be a manifold of $m$--dimensional, and $M\subset \mathbb{R}^k$. Assume $m>n$. If every smooth function $f:M\longrightarrow \mathbb{R}^n$ has regular values that form an open subset of ...
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64 views

show that $R^n -X$ has at most 2 connected components.

show that $R^n -X$ has at most 2 connected components. Theprem: Suppose that $X$ is the boundary of $D$, a compact manifold with boundary and let $F:D \to R^n$ be a smooth map extending $f$; ...
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every $f \in F^k_p$ has a Taylor expansion

$F_p$ is the set of germs of functions on a manifold M which vanish at $p \in M$. Let $F^k_p$ be the ideal of $C^\infty(p)$ generated by $f_1,... \,f_k$, where $f_i \in F_p$. (i.e $F^k_p$ is $\sum ...
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contour which can be homeomorphic?

If I have a function $\phi:\mathbb{R^{2}}\rightarrow\mathbb{R}$ which is $C^{\infty}$ without critical points, can I assure that all the contour are homeomorphic?
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Wedge product = set intersection?

In a research article [1] I found the following formulation: The wedge product may be considered as set intersection. For example, surfaces of constant $f(x,y,z)$ and surface of constant ...
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125 views

Diffeomorphism from level set onto $S^2$

I'm given a map $\phi : R^4 \rightarrow R^2$ defined by $\phi(x,y,s,t) = (x^2 + y, x^2+y^2+s^2+t^2+y)$. It's easy to show that the level set $C = \phi^{-1}(0,1)$ is a smooth submanifold of $R^4$ with ...
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Are $C^\infty$ exotic spheres $C^k$ exotic?

The only theory of exotic spheres that I know is of $C^\infty$ structures on them; that is, that there are plenty of spheres (in dimensions $n \geq 7$ that are homeomorphic but not diffeomorphic. To ...
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Show that there exist a compact manifold with boundary $W$ in $Y \times I$ such that $\partial W= X\times \{0\} \cup Z \times \{1\}$

I found this question had been posted by someone, but got no answer, so I hope I will have better luck. Show that if $X$ may be deformed into $Z$ then $X$ and $Z$ are cobordant. Deformation ...
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$\mathbb{C}P^1$ diffeomorphic to $S^2$

I am trying to show that the complex projective line is diffeomorphic to the 2-sphere. I'm using the $C^{\infty}$ structure on $\mathbb{C}P^1$ given by $U_1 = \{ [z_1 : 1], z_1 \in \mathbb{C} \}$, ...
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106 views

Lie algebras of vector fields over $\mathbb{R}$ and over $S^1$

An exercise in the book "Moonshine beyond the Monster" has me stumped. It asks whether the real Lie algebras of smooth vector fields over the reals $V(\mathbb{R})$ and over the circle $V(S^1)$ are ...
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76 views

Using the results of the local immersion/submersion theorems on manifolds

When $X,Y$ are $k$- and $l$-manifolds, we can have a function $f:X\rightarrow Y, x\in X$ such that $f$ is an immersion resp. submersion at $x$. The local immersion/submersion theorem now says: There ...
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Intuition about Morse functions

We have defined: a Morse function on $X$ is a smooth function $f:X\rightarrow\mathbb R$ with only non-degenerate critical values. I tried to get some intuition about this, and found the section Basic ...
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isotopy of homeomorphisms of a torus

Let's consider some homeomorphism of a torus which is isotopic to identity. Is it possible to construct an explicit isotopy? Edit: It's well-known statement that a homoemorphism of a torus is ...