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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Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
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3answers
46 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
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4answers
531 views

Why is there no natural metric on manifolds?

One of the things that always bothered me after learning introductory differential geometry (as a physics student) and then delving deeper into this field on my own is that, the usual construction of ...
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127 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
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83 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
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34 views

Why does every noncompact orientable surface have a complex structure?

There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ...
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1answer
59 views

Are definitions for smooth map between manifolds are equivalent?

There are two ways to define a smooth map between manifolds. The 1st way (for example, Lee): $f:M\rightarrow N$ is smooth iff for every $p\in M$ there exist charts $(U,\varphi)$ at $p$ and $(V,\psi)$ ...
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44 views

How does a differential act when we identify $T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N$?

It's fairly common to identify the tangent space of a product manifold as $$ T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N $$ where $p=(p_1,p_2)$, and the actual isomorphism is given by $v\in ...
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33 views

If two Curves in $\Bbb R^3$ are transversal then they do not intersect

Proving this is easy: Let $X$ and $Z$ be the two curves in $\Bbb R^3$. Assume $X$ and $Z$ intersect at a point $y$. Then, at most $\dim(T_y(X) + T_y(Z)) = 2$, where $T_x(X)$ and $T_z(Z)$ denotes the ...
2
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64 views

If $S\subseteq M\times N$ is embedded, and $S$ and $\{p\}\times N$ intersect transversely in one point, then $\pi_M|_S$ is a diffeomorphism?

I'm trying to prove the equivalence of the following statements: Suppose $M^m$ and $N^n$ are smooth manifolds, $S\subseteq M\times N$ immersed, and $\pi_M$ and $\pi_N$ the projection maps. TFAE: ...
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31 views

Transition Functions for Cartesian Coordinate Systems

This is my first time using Mathematics SE (I've only used Physics and Astronomy before), so I apologize if this question is awkwardly phrased or incorrectly presented. I welcome any and all edits and ...
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2answers
60 views

Proving that a form is exact

Maybe this question is rather obvious but I didn't manage to solve it myself. Assume $M$ is a closed, oriented manifold. take $$ \Omega^k(M)\ni \omega = \begin{cases} d\beta~,~~~ in~ U\\ 0~,~~~ ...
2
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1answer
30 views

Does Heine-Borel hold for smooth manifolds?

If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$. Does the Heine-Borel theorem still ...
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92 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
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1answer
38 views

Why are the basis elements of $T_pM$ a vector field if we let $p$ vary?

At the end of this article on tangent vectors, they say each $\frac{\partial}{\partial x_i}$ is a vector field, if we let the point $p$ vary. However, they are only defined in a particular coordinate ...
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1answer
26 views

If $v\in\mathbb{R}^N\setminus\mathbb{R}^{N-1}$, what is the projection $\pi_v$ with kernel $\mathbb{R}v$?

I going over a lemma for the Whitney Embedding theorem which shows that an injective immersion of an $n$-manifold into $\mathbb{R}^N$ can actually be immersed in a lesser dimensional Euclidean space ...
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1answer
52 views

If $b$ is a regular value of $f$, $f^{-1}(-\infty,b]$ is a regular domain?

I'm trying to prove the first part of Proposition 5.47 of Lee's Smooth Manifolds, which is left to the reader. It says Suppose $M^m$ is a smooth manifold, and $f\colon M\to\mathbb{R}$ smooth. For ...
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1answer
34 views

Show that a set is a smooth curve and find a parameterization for it.

Let $S = \{ (x,y,z) \in \mathbb{R}^3 \mid x - yz + z^3 = 0 \}$. Let $\pi: \mathbb{R}^3 \to \mathbb{R}^2$ be such that $\pi(x,y,z) = (x,y)$. Let $H = \{p \in S \mid \pi_{\mid S}: S \to \mathbb{R}^2 ...
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1answer
69 views

Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth ...
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36 views

About $C^{0}$ being topological manifold

Is that the reason why $C^{0}$ being topological manifold due to that $C^{0}=\phi$ which contains nothing? Correct me if I am wrong. I am new to differential topology.
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48 views

Is $y^2=x(x-1)^2$ an immersed submanifold?

Is the curve $y^2=x(x-1)^2$ an immersed submanifold in $\mathbb{R}^2$? It's certainly not embedded since it intersects itself at $(1,0)$. I'm aware of techniques to prove a subset is not a immersed ...
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2answers
39 views

Why is the irrational winding of the torus not locally path connected?

The irrational winding of the torus given by the map $f\colon\mathbb{R}\to T^2$ where $f(t)=(e^{it},e^{i\alpha t})$ for some irrational $\alpha$. Wikipedia mentions this is not a regular submanifold, ...
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85 views

Is $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ diffeomorphic to $S^2$?

I was working on Problem 5-1 of Smooth Manifolds by Professor John Lee, and it lead me to wanting to show that $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ is diffeomorphic to $S^2$, and that is ...
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1answer
38 views

Smooth embeddings of the $2$-sphere

I have a past qual question here: given a smooth embedding $f \colon S^2 \to \mathbb{R}^3$, show that there must exist distinct points $p,q \in S^2$ such that the tangent planes to the embedded sphere ...
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1answer
58 views

Video/audio lectures on differential topology?

Do there exist decent online video lectures, or even audio lectures, covering differential topology? I'm aware of Milnor's talk, but it is more like exposition and doesn't go very far.
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1answer
41 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
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17 views

Is there a rule for computing the differential of a product of maps?

A lot of partition of unity arguments have some map of the form $f=\sum_i \psi_if_i$. Is there a formula for the differential $df_p$ in terms of its summands? For instance, suppose $f_i:U_i\to V_i$ ...
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2answers
27 views

Inward and outward pointing tangent vectors?

If $M^n$ is a smooth manifold with boundary and $p\in\partial M$, then $T_pM$ is the disjoint union of inward and outward point vectors, and $T_p\partial M$. If $(U,(x^i))$ is a smooth boundary chart ...
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52 views

How does $v\Phi^1=\cdots=v\Phi^k=0$ imply $v\in\ker d\Phi_p$?

I'm confused about an immediate corollary in John Lee's Smooth Manifolds. Proposition 5.38 says Suppose $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold. If $\Phi\colon U\to N$ ...
2
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1answer
43 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
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0answers
25 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
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11 views

Understanding topological modular forms

I am asking for good references to understanding topological modular forms. Please don't laugh. I am more or less an analyst and differential geometer, and I do know some algebraic topology and very ...
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1answer
39 views

Why does $S^n$ satisfy the local $n$-slice condition? From Lee's Smooth Manifolds.

Example 5.9 on page 103 of Lee's Smooth Manifolds says the following: The intersection of $S^n$ with the open subset $\{x:x^i>0\}$ is the graph of the smooth function $$ ...
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58 views

Voisin's proof of Ehresmann's theorem

On p.221 of Voisin's book on Hodge theory, there are two claims: a) Let $B$ be a contractible smooth manifold. There exists a vector field $\chi$ on $B$ whose flow $\Phi_t$ is global and, given any ...
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1answer
34 views

Why is the inverse of the standard charts on $\mathbb{R}P^n$ continuous?

When showing that $\mathbb{R}P^n$ is a topological manifold, the atlas is given by charts $\varphi_i\colon U_i\to\mathbb{R}^n$, where the $U_i$ are the classes $[x^1,\dots,x^{n+1}]$ with $x_i\neq 0$, ...
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32 views

A question about the boundary points of manifolds.

A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$. The "centred at ...
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46 views

Pontryagin class of 0-manifolds

If $f:M^m \rightarrow S^m$ (where $M$ is a smooth, connected and orientable manifold), is a smooth map and $y \in S^m$ is a regular value of $f$, then $f^{-1}(y)$ is a framed $0$-manifold, that is, ...
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1answer
35 views

Condition for Orientability of Manifold

Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable. $M$ is orientable iff all ...
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1answer
40 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
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1answer
48 views

Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$

i.e. Does $\mathbb{R}P^n$ have a tubular neighborhood $N$ such that $N-\mathbb{R}P^n$ is disconnected. My guess is yes, but don't know how to show it convincingly ( or maybe only for $n$ odd, I'm ...
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1answer
40 views

Vanishing pushforward implies smooth function is locally constant?

I'm trying to prove that if the pushforward $dF$ of a smooth map $F\colon M\to N$ between smooth manifolds is zero, then $F$ is constant on each component. It will be enough to show $F$ is locally ...
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1answer
23 views

Inverse Image of a Regular Value an Orientable Submanifold

Let $f:M^n \rightarrow \mathbb{R}$ be a smooth map, and let $c\in N$ be a regular value. When is $f^{-1}(c)$ an orientable manifold? Note: I know by regular value thm, $f^{-1}(c)$ is a smooth $n-1$ ...
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3answers
247 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
3
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1answer
61 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
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50 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
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1answer
45 views

Is a level set of a manifold a set of zeroes

Suppose $X$ and $Y$ are manifolds of dimensions $k$ and $l$ (with $k>l$). Given $F : X \to Y$ a smooth map and $y$ a regular value in $Y$, does there exist a map $G : X \to \mathbb{R}^l$ such that ...
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2answers
125 views

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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191 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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129 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 ...
8
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1answer
159 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 ...