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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Differential forms, number of zeros on disk

Let $f$ be a holomorphic function on $U \subset \mathbb{C}$ and $f'(z) \neq 0$, $z \in U$. Then how do I show that all zeros of $f$ are simple and positive, and that if I have a disk $D \subset U$ ...
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134 views

Pullback of a 1 form on the circle

Q: Let $M$ be a smooth compact manifold, and suppose there is a smooth map $F:M \rightarrow S^{1}$ whose derivative is non-zero at every point. Prove that the de Rham cohomology space $H^{1}(M)$ is ...
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72 views

Eembedding of product $\mathbb{S}^2\times\mathbb{S}^3$ into $\mathbb{R}^6$

It is easy to see that $\mathbb{S}^n$ can be embedded in $\mathbb{R}^{n+1}$ and therefore $\mathbb{S}^2\times\mathbb{S}^3$ can be embedded in $\mathbb{R}^7$. The question is how to prove that ...
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167 views

How does the Möbius group act on circlines?

This is a continuation of my earlier, rather vague question. I am interested in studying the action of the Möbius group $PGL(2,\mathbb{C})$, on the circlines in the extended complex plane $\mathbb{C} ...
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87 views

How do you work with the space of circles on the sphere considered as the projective line?

I'm trying to prove some things about the action of the Möbius group on the "circlines" in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius ...
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37 views

Prove that in the half plane $\{x>0\}$, ω is the differential of a function.

Define a 1-form $ω$ on the punctured plane $R^2-\{0\}$ by $ω(x,y)=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2 }dy $ a) Calculate $∫_Cω$ for any circle C of radius r around the origin b) Prove that in ...
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136 views

cohomology homomorphism between grassmannians induced by inclusion

Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians. Then $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
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79 views

Show that the volume element of $V$ is $ϕ_1\wedge\cdots\wedge ϕ_k$.

a) Let $V$ be an oriented $k$-dimensional vector subspace of $\mathbb{R}^N.\,$Prove that there is an alternating $k$-tensor $T\in\bigwedge^k (V^*)$ such that $T(v_1,\ldots,v_k )=1/k!$ for all ...
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Bases $\{v_1,…,v_k \}$ and $\{ v_1' ,…,v_k' \}$ for $V$ are equivalent iff $T(v_1,…,v_k )$ and $T(v_1',…,v_k' )$ have the same sign

a) Let $T$ be a non zero element of $∧^k (V^*)$ where $\dim⁡ V=k$. Prove that 2 ordered bases $\{v_1,…,v_k \}$ and $\{ v_1' ,…,v_k' \}$ for $V$ are equivalent oriented if and only if $T(v_1,…,v_k )$ ...
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cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
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176 views

Embedding a torus in $\mathbb{R}^{n+1}$

It is easy to see that $T^2=S^1 \times S^1$ can be embedded in $\mathbb{R}^4$ but also there is an embedded in $\mathbb{R}^3$. The question is $T^n = S^1 \times \ldots \times S^1$ can be embedded ...
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1answer
130 views

About simple connectedness

Two topological spaces $X$ and $Y$ are homotopic if there exists continuous $f: X \to Y$ and $g: Y\to X$ such that $f\circ g$ is homotopic to $Id_Y$ and $g\circ f $ homotopic to $Id_X$ (regardless any ...
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33 views

How to show $g^{-1}\circ f:\partial N\longrightarrow \partial N$ extends to a diffeomorphism $h:N\longrightarrow N$?

Let $M$ and $N$ be two manifolds with boundary and let $$f, g:\partial N\longrightarrow \partial M,$$ two isotopic diffeomorphisms, that is, there exists a diffeomorphism $$F:\partial N\times [0, ...
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1answer
207 views

Milnor's proof of the fundamental theorem of algebra (Topology from the Differentiable Viewpoint)

I am studying the proof of the fundamental theorem of algebra out of John Milnor's book Topology from the Differentiable Viewpoint, located on page 8 here: ...
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78 views

Transversality through two functions $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ for $W\subset Z$

For Exercise 5 section 5 chapter 2 of Guillemin & Pollak: Set $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ and assume that g is transversal to a submanifold $W\subset Z$. Show $f\pitchfork g^{-1}(W)$ ...
2
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138 views

Prove that nondegenerate zeros are isolated.

a) Prove that nondegenerate zeros are isolated. b) Furthermore, show that at a nondegenrate zero $x$, $ind_x (\vec v)=+1$ if the isomorphism $d(\vec v_x )$ preserve orientation, and $ind_x (\vec ...
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1answer
55 views

What are the charts that make up an atlas for the long line?

This question is prompted while I was working through the MIT OCW (Massachusetts Institute of Technology, Open CourseWare) for 18.965, ``Geometry of Manifolds,'' in its Lecture 2, ...
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26 views

Show that at a zero $x$, the derivative $d(\vec v_x ) :T_x (X)→R^N$ actually carries $T_x (X)$ into itself.

Show that at a zero $x$, the derivative $d(\vec v_x ) :T_x (X)→R^N$ actually carries $T_x (X)$ into itself. I know that we have the vector field $\vec v:X\to R^n$. If $X=R^n \times \{0\}$, then the ...
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62 views

Why aren't those spaces diffeomorphic?

(Taken from Bredon - Topology and Geometry): Let $X$ be the graph of the real valued function $\theta(x)=|x|$ of a real variable $x$. Define a functional structure on $X$ by taking $f \in F(U) ...
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96 views

Why the map $z→z+ \overline z^m $ has fixed point with local Lefschetz number $m$ at the the origin of C (m≥0)?

My professor went through an example in class and making following claim The map $z→z+z^m$ has a fixed point with local Letfschetz number $m$ at the origin of $C$ $(m>0)$ For any $c≠0$, the ...
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93 views

fibration of real projective space over complex projective space

From the fibration $$U(1)=S^1\to S^{2n+1}\to \mathbb{C}P^n,$$ can we quotient the action of $S^0$ and obtain a well-defined fibration $$ S^1/S^0=\mathbb{R}P^1\cong S^1\to ...
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1answer
87 views

fibration of complex projective space over quaternion projective space

From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
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Prove that the diagonal $∆$ in $S^2×S^2$ is not globally definable by $2$ independent function.

Prove that the diagonal $∆$ in $S^2×S^2$ is not globally definable by $2$ independent function. In contrast, show that the other standard copies of $S^2$ in $S^2×S^2$ – ie, $S^2×\{a\}$ for $a∈S^2$ are ...
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63 views

Comparison between two definitions of real projective spaces.

The most common definitions of real projective spaces are: $\mathbb{R} \mathbb{P} ^n = (\mathbb{R}^{n+1} - 0)/ \sim$, where $x,y \in \mathbb{R}^{n+1}-0$ satisfies $x \sim y$ iff $x = \lambda y$ for ...
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mathematical limit for a ouroboros torus

The other day i was watching an episode of Tom and Jerry in which a similar situation was present toms head comes out of his own mouth. My head hurts when i think how is that even possible so i ...
3
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1answer
78 views

Is the disjoint union of submanifolds a submanifold?

Let $M$ a manifold and $X \subset M$. Let $N \subset X$ such that $N, X \backslash N$ are submanifolds of $M$. Can I conclude that $X$ is a submanifold of $M$?
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37 views

Restricting a differentiable function to a submanifold.

If $f: M \longrightarrow N$ is a differentiable function between manifolds and $A$ is a submanifold of $M$, can I conclude that $f_{|_A}$ is a differentiable function? It seems that the answer should ...
2
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1answer
183 views

Torus/Möbius Band homeomorphism

Is a fattened Möbius Spiral Band homeomorphic to a Torus? (Due to the same Euler Characteristic $\chi$ ?) Are both non-orientable? Following (3D printable plastic) Torus has a square section that ...
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Boundary of $\mathbb{R}^4$ and fundamental group of $\mathbb{R}^4/\mathbb{R}^2$

a) To what I know boundary makes no sense in open sets. Does it make any sense to talk about the boundary of $\mathbb{R}^4$? In physics they consider it when discussing wether the universe has a ...
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1answer
39 views

What is $C_n(X)$?

This article says the following: Let $X$ be a triangulated space and let $C_n(X)$ be a real vector space with $n$-simplices $[x_0,x_1,x_2,\dots,x_n]$. Each different combination of $x_i's$ forms a ...
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Definition of a Manifold from Guillemin Pollack

I have been studying differential topology from Guillemin and Pollack (GP). Unlike many other books that define differentiable manifolds using maximal atlases GP starts by saying $ X \subset R^{N}$ ...
2
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1answer
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Equivalent definitions of Euler characteristic for closed manifolds

It is well-known that the Euler characteristic of a closed manifold $M^n$, which can be defined as $\chi(M)=\sum_{k=0}^n (-1)^k \operatorname{dim}H^k(M)$, equals the intersection number ...
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Show that $R^n -X$ has at most 2 connected component without using the Jordan-Brouwer theorem.

Here is what I got but my professor say it's wrong Let $X$ be a compact, connected hyperspace in $R^n$, then $ R^n-X$ consist of 2 open sets $D_0$ – the outside component and $D_1$ the inside ...
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1answer
69 views

Basis for a line bundle

I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text: "Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, ...
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1answer
172 views

Prove directly Morse lemma for real line $\mathbb{R}$

The exercise 9 section 7 chapter 1 in Guillemin & Pollak state the next Prove directly Morse lemma for real line $\mathbb{R}$.(Hint:Use this elementary calculus lemma: for any function on ...
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69 views

Lie group quotient structure

Let $G$ be a Lie group and $H$ a normal finite subgroup. Let $\pi : G \to G/H$ be the quotient surjection. How would one show that $G/H$ is a Lie group?
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Subsets of a manifold.

I read that every open subset $A$ of a manifold $M$ is a submanifold (it is a manifold with the induced topology by $M$). If I understand correctly, the argument is that, for an element $x \in A$, one ...
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Extenting a homeomorphism on subsurface to the entire surface

Suppose you have surface and a subsurface. The complement of subsurface is union of open discs and once punctured open discs. Can all homeomorphisms of the subsurface be extended to homeomorphisms of ...
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Preimage of 0 for a differentiable function.

If a subset $N$ of a manifold $M$ can be written as $f^{-1}(\{0\})$ being $f:M \longrightarrow \mathbb{R}$ a differentiable function, can I conclude that $N$ is a submanifold of $M$?
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How to prove that a certain set is a submanifold.

Let $P^{n-1}(\mathbb{R})$ the real projective space of dimension $n-1$. Consider the set $$B=\{(x,y)\in\mathbb{R}^n \times P^{n-1}(\mathbb{R}) / x=(x_1,..,x_N), y=[y_1;..;y_N], x_iy_j=x_jy_i ...
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1answer
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Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}}) $ (open ...
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1answer
38 views

1-forms as a direct sum

Let us define the spaces $C_0, C_1$ and $C_2$ of differential $0,1,2$ forms respectively on the sphere $S^2.$ Is it true that $C_1$ is the direct sum of $d(C_0) \oplus \delta^* (C_2)$? I think this ...
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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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52 views

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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185 views

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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70 views

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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151 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 ...
0
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1answer
49 views

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 ...
4
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2answers
218 views

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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1answer
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 ...