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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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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37 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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57 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
108 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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101 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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42 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
82 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
52 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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19 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
45 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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55 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 $\Phi\colon U\to N$ ...
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1answer
76 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
32 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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14 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
50 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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64 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
36 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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37 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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49 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
38 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
48 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
53 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
74 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
33 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
345 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?
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1answer
64 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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0answers
56 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
47 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
131 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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2answers
223 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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134 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
184 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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23 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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1answer
28 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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86 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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1answer
33 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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67 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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0answers
44 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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2answers
59 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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2answers
122 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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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 $\mathrm{supp}\tilde{f}=\overline{\bigcup_{p\in A}\mathrm{supp}\psi_p}$. ...
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3answers
124 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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0answers
68 views

Alexandrov embedness and branch points

Let assume that $\Sigma_n$ is a sequence of compact surfaces in $\mathbb{R}^3$ of fixed genus. We assume that the surfaces are Alexandrov embedded, that is to say there exits an immersion $i_n$ from ...
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4answers
362 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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0answers
78 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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2answers
65 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
50 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
40 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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0answers
19 views

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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1answer
90 views

Is complex projective space simply connected?

I know real projective space isn't simply connected, what about complex projective spaces?