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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Non existence of a non singular vector field on $S^2$

Prove that the unit tangent bundle of $S^2$, $T^1 S^2$, is not diffeomorphic to $S^2×S^1$ by showing that if so there exists a nowhere vanishing vector field on $S^2$ I do not know how to create that ...
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74 views

Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
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21 views

Differential of rotation matrix at the north pole of sphere

Let T(p) rotate $p\in S^{2}$ by angle $\theta $ about the z-axis. The problem is to compute $dT_{(0,0,1)}$. T can be represented by the usual 3x3 rotation matrix $A_{z}(\theta)$. So $T(p)=A_{z}p$. ...
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1answer
75 views

Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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43 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
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1answer
118 views

Does de Rham theorem hold for manifolds with boundary?

I am following the J.Lee's book "Introduction to Smooth Manifolds", 2nd ed., page 480-486 to learn the de Rham theorem. It is proven on manifolds without boundary, which makes me curious about whether ...
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18 views

Show differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ is injective

The problem is find to the differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ (spheres) defined as $f(x_{0},...,x_{n},y_{1},...y_{n})=(x_{0}y_{0},x_{0}y_{1},...,x_{n}y_{1},x_{n}y_{n})$ and show it is ...
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54 views

English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of Dieudonné's Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
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44 views

Is there a characteristic property of quotient maps for smooth maps?

If $\pi\colon X\to Y$ is a quotient map, and $f\colon Y\to Z$ is a continuous map between topological spaces, then the characteristic property of the quotient map says $f$ is continuous iff $f\circ ...
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2answers
95 views

Stiefel-Whitney classes of 3-manifolds are trivial

Is there a simple way how to show that Stiefel-Whitney classes of a compact closed 3-manifold $M$ are zero? This is exercise 11-D in Milnors Characteristic classes. The available tools in the ...
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Zeros of vectorial field [on hold]

Given a $M$ manifold in ${\mathbb R}^n$ and $X:M\rightarrow TM$ a vectorial field such that $\pi\circ X=Id$ where $\pi:TM\rightarrow M$ (projection to $M$). One zero of $X$ is such that ...
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59 views

Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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1answer
26 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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36 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
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39 views

Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
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65 views

Why Must the Degree of this Map be 0? [on hold]

Let $f:S^3 \rightarrow S^1\times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$. (Just a hint would be good)
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72 views

How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
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2answers
29 views

Equality between support for a function and closed union of elements of a partition of unity (Proof from John Lee's Smooth Manifolds)?

I have a minor question in the following proof from John Lee's Intro to Smooth Manifolds: At the end, there is the equality $\mathrm{supp}\tilde{f}=\overline{\bigcup_{p\in A}\mathrm{supp}\psi_p}$. ...
2
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2answers
63 views

Embedding of $\mathbb{R}^2 \to \mathbb{R}^3$ with non-parallel tangent planes

I have a qual question here and I'm struggling to get a good starting point. The question asks to construct a smooth proper embedding $f\colon \mathbb{R}^2 \to \mathbb{R}^3$ such that for any distinct ...
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1answer
243 views

Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
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4answers
288 views

Should I read about Manifolds or Algebraic Topology?

I really enjoy doing maths and it fills quite a lot of my spare time. I'm starting my first year in the university on october and I probably won't have that much time for independent reading once ...
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45 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
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1answer
45 views

Smooth Submanifolds of $\mathbb{RP}^3$

Let $ M=\{[z_0,z_1,z_2, z_3] \in \mathbb{RP}^3 | (z_0-z_3)^2+az_1^2=0\}$, where $a\in \mathbb{R}$. Show that $M$ is a smooth submanifold of $\mathbb{RP}^3$ of dimension $2$ when $a=0$, but not if ...
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1answer
35 views

Show this Map $f:M\rightarrow S^n$ is a Diffeomorphism

Let $M$ be a compact, connected smooth manifold of dimension $n>1$, and $f:M\rightarrow S^n$ a smooth map such that the differential $df_x:T_xM\rightarrow T_{f(x)}S^{n}$ has full rank $= n$ for all ...
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1answer
63 views

Is complex projective space simply connected?

I know real projective space isn't simply connected, what about complex projective spaces?
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Is the natural map $\Omega^*(M)\otimes \Omega^*(F) \rightarrow \Omega^*(M\times F)$ injective?

As I self-study Bott and Tu 's Differential Forms in Algebraic Topology, I am stopped by some quite basic questions: 1) Let $M$ and $F$ are real manifolds, and let $\pi : M\times F \rightarrow M$ and ...
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117 views

Avoiding vertical vectors in tangent spaces.

Let $i: [0,1]\hookrightarrow\mathbb{R}^3$ be a smooth embedding. Can I find arbitrary small perturbations $j$ of $i$ s.t. $j(0)=i(0)$, $j(1)=i(1)$ and $j'(t)$ is never a multiple of $(0,0,1)$? More ...
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1answer
59 views

Derivative of the inclusion of a submanifold

I know there are other questions similar to this one, but I just want you to tell me if what I'm doing is rigth and how to improve it. The problem is the following: (I'm using the definitions by ...
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1answer
30 views

Two definitions of framed manifolds/cobordism

One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact ...
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1answer
70 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
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1answer
404 views

Different definitions for submanifolds

I'm trying to better understand the concept of differentiable submanifold. However, it looks like many different definitions are adopted by various authors and so I'm trying to keep myself in sync by ...
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1k views

Is every embedded submanifold globally a level set?

It's a well-known theorem (Corollary 8.10 in Lee Smooth) that given a smooth map of manifolds $\phi:M\rightarrow N$ and a regular value $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a ...
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3answers
84 views

Simply connected manifolds are orientable

For a simply connected $n$-manifold $M\subseteq\Bbb{R}^k$, I want to show that $M$ is orientable. Take a point $p\in M$ and take an $n$-disc, $D^n$, around $p$ (we can take it as small as we please). ...
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217 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
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58 views

Preimages of smooth maps between manifolds

Suppose $M$ and $N$ are both compact, connected, oriented $m$-manifolds without boundary and $f: M \to N$ is smooth. What additional condition(s) must $f$ satisfy so that there exists at least one ...
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32 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
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1answer
50 views

Definition of “Representing” a Handlebody (Lefschetz Fibration)?

Sorry, I could not find a clear explanation of the meaning of the word represented in the following:"any 4-dimensional 2-handlebody W can be represented by a topological (achiral) Lefschetz fibration ...
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54 views

Almost Every Hyperplane is Transverse to $M$

Let $M$ be an $n$-dimensional manifold embedded in $\mathbb{R}^{n+1}$. I am trying to show that almost every hyperplane in $\mathbb{R}^{n+1}$ is transverse to $M$. To show that I would like to prove ...
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How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
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3answers
125 views

Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
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Parametrizing Walks on Sphere and Torus

This question is very underdeveloped, but I was wondering if there was a map from the sphere to the torus which preserves length of closed curves? I was just thinking about taking a walk on a ...
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Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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Generalizations of the product neighborhood theorem

As far as I know, the Product Neighborhood Theorem for smooth manifolds states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization ...
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1answer
56 views

Proving a Certain Smooth Map $S^n\rightarrow S^n$ is a Diffeommorphism

I am given a smooth map $f:S^n\rightarrow S^n$, for $n\geq 2$, whose differential is injective at each point. I am asked to prove that it is a diffeomorphism. Since the differential is injective ...
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2answers
85 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
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1answer
58 views

Show That $\dim H_m(\partial M;\mathbb{R})$ is Even

A student asked me this. Suppose that $M$ is a compact, orientable $n$-manifold with boundary. It is a fact that for each $k$ with $0\leq k\leq n$ the vector spaces $H_k(M;\mathbb{R})$ and ...
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2answers
372 views

Product of spheres embeds in Euclidean space of 1 dimension higher

This problem was given to me by a friend: Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$. The solution is apparently ...
8
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1answer
209 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
14
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1answer
524 views

Compactification of Manifolds

It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X. In the case of the (differentiable) manifold $\mathbb R^n$ this one-point ...
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a question related with morse theory [duplicate]

Show that there exists no smooth function $f:\mathbb{R}^2→\mathbb{R}$,such that $f(x,y)\geq 0$ for any $(x,y)\in\mathbb{R}^2$, with exactly two critical points$(x_1,y_1)\in\mathbb{R}^2$, ...