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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15
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66 views

group operations are smooth in $\text{SL}(n, \mathbb{R})$

I am told the following reason as to why group operations of multiplication and inversion are smooth on $\text{SL}(n, \mathbb{R})$. Multiplication is smooth because the matrix entries of a product ...
2
votes
2answers
54 views

Immersion of $\mathbb{R}$ to $\mathbb{R}^2$

I have no idea how to prove that the set $\{(x, |x|): x\in\mathbb{R}\}$ is not the image of an immersion of $\mathbb{R}$ into $\mathbb{R}^2$ For example If $f(t)=(t^3, |t|^3)$ then $f'(0)=(0, 0)$.
7
votes
1answer
32 views

defining $C^\infty$ structure on finite-dim vector space, homeomeomorphism to tangent bundle, such that independent of choice of bases

If $V$ is a finite dimensional vector space over $\mathbb{R}$, how would I go about defining a $C^\infty$ structure on $V$ and a homeomorphism from $V \times V$ to $TV$ which is independent of bases? ...
12
votes
3answers
243 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
0
votes
1answer
18 views

something about diffeomorphism

Suppose $A$ and $B$ are both open sets, and there is a diffeomorphism $g$ between them. My book says that the chain rule implies that $Dg$ is non-singular. I don't understand. Can anyone tell my why?
8
votes
2answers
98 views

Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
6
votes
2answers
91 views

Manifold Orientability Definition

In Shigeyuki Morita's Geometry of Differential Forms, orientability is defined in the following way: If we can assign an orientation to each point on a manifold $M$ in such a way that the ...
6
votes
1answer
66 views

Is the Whitney embedding theorem tight for all $n$?

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. In dimension 1 this is tight: the circle cannot be embedded into $\Bbb R^1$. It is a ...
5
votes
1answer
41 views

identity map is not diffeomorphism, $x^3$ is a diffeomorphism [closed]

Consider the real line $\mathbb{R}$ the two following differentiable structures: 1) $(\mathbb{R}, f_1)$ where $f_1(x) = x$. 2) $(\mathbb{R}, f_2)$, where $f_2(x) = x^3$. How do I demonstrate that: ...
0
votes
0answers
26 views

Index of a non-degenerate critical point of a smooth holomorphic function

I am looking for hints, and ideally for several ways to approach the following problem. Consider a holomorphic function $\mathbf C^n \to \mathbf C$, that is smooth (we view $\mathbf C^n$ as a ...
4
votes
1answer
24 views

What can you say about injection, immersion, embedding for the torus?

Define $\varphi_a: \mathbb{R} \to T$ where $T = S^1 \times S^1$ is the torus via$$\varphi_a(x) = (e(x), e(ax)),\text{ }e(x):=e^{2\pi i x}$$and $a > 0$ is some parameter. Determine for which $a$ ...
1
vote
2answers
20 views

Possibility of the cellular decomposition of a manifold

I am working to find if it's possible to find a cellular decomposition of $S^2\times S^1$ as following: $e^0\cup e^1\cup e_1^2\cup e_2^2\cup e^3$. I cannot find such a decomposition. And I try to ...
3
votes
1answer
80 views

Non homeomorphism

I want to show that the sphere $S^2$ and the torus $T^2$ are not homeomorphic, using the notion of intersection modulo $2$. I have to show that any two loops on the sphere $S^2$ have an even number of ...
2
votes
1answer
38 views

Smooth function on intersection is the difference of two smooth functions

I am trying to understand a proof from Loring W. Tu's An introduction to Manifolds. In order to prove Proposition 26.2, The author must show that if $\{U, V\}$ is a open cover of a manifold $M$ and ...
3
votes
1answer
25 views

definition of differential does not depend on the choice of the curve, differential is a linear map

Let $\varphi: M \to N$ be a differentiable map. How do I show that the definition of the differential $d\varphi_p: T_pM \to T_{\varphi(p)}N$ of $\varphi$ at $p$ does not depend on the choice of the ...
3
votes
1answer
30 views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of ...
1
vote
1answer
85 views

Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even

I want to prove: Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even. I'm stuck. My thoughts are: I want to use the hairy ball theorem and I want to ...
1
vote
0answers
15 views

Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
4
votes
1answer
22 views

no point where assumptions of Inverse Function Theorem hold?

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}$, $\psi: \mathbb{R}^3 \to \mathbb{R}$ are continuously differentiable. Define $f: \mathbb{R}^3 \to \mathbb{R}^3$ by$$f({\bf x}) = (\varphi({\bf x}), \psi({\bf ...
6
votes
1answer
119 views

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 $D \subset U$ ...
2
votes
0answers
39 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 ...
1
vote
1answer
46 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 ...
2
votes
1answer
63 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} ...
3
votes
1answer
56 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 ...
1
vote
0answers
24 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 ...
0
votes
0answers
28 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 $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
0
votes
1answer
55 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 ...
0
votes
1answer
50 views

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 $T(v_1,…,v_k )$ ...
2
votes
0answers
30 views

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 ...
6
votes
2answers
105 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 ...
1
vote
1answer
100 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 ...
0
votes
0answers
24 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, ...
0
votes
1answer
55 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: ...
1
vote
1answer
29 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
votes
1answer
55 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 ...
1
vote
1answer
25 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, ...
1
vote
0answers
18 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 ...
1
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0answers
38 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) ...
1
vote
1answer
57 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 ...
2
votes
1answer
41 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 ...
1
vote
1answer
26 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 $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
5
votes
2answers
97 views

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 ...
2
votes
1answer
27 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 ...
2
votes
0answers
33 views

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 ...
0
votes
0answers
24 views

Exhaustion of a manifold by compacts

I searched for a proof of the following statement, but did not find one. I want to check if a proof I made is correct, or if I'm leaving out some detail and/or complicating things: Proposition: ...
1
vote
1answer
24 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 $M$?
1
vote
0answers
25 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 ...
0
votes
0answers
13 views

If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that if $s(p,q) =0$ then $$ \nabla ...
1
vote
1answer
64 views

Torus/Moebius Band homeomorphism

Is a fattened Moebius 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 ...
0
votes
3answers
75 views

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 ...