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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The derivative of the inclusion map is the inclusion map of tangent spaces.

Let $X$ and $Y$ be smooth manifolds, let $i:X\to Y$ be the inclusion map, prove $di_x$ is the inclusion map from $T_x(X)$ to $T_x(Y)$. I know this is pretty basic, but can someone show me how to do ...
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Why is a vector space equal to its tangent space for any point?

I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$. If $x\in V$, then since $V$ ...
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The space of paths

Let $ (M,g) $ be a compact Riemannian manifold, and let the path space $ \Omega $ of $ M $ be the set of equivalence class of smooth maps $ \gamma : [0,1] \rightarrow M $ (equivalent under ...
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Defining a quotient manifold with gluing

I'm trying to find conditions on the gluing map between two manifolds so that the quotient space will be a smooth manifold, and the inclusion map will be a diffeomorphism. Specifically, Suppose ...
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Functions from $[E_8]$ to $E_8$

Let $f: [E_8] \to [E_8]$ be a function between 4-manifolds with intersection form $E_8$. What we know (due to Rocklin) is that $[E_8]$ can't have any smooth structure. Questions: Is it true for all ...
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Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector ...
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91 views

Is this function just zero?

I asked a question like this a few minutes, but am about ready to strike the problem out as errata for the book. The problem defines a function $g(x)=f(x-a)g(b-x)$. The function has bound $|f|<1$, ...
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Does a germ of a smooth (i.e., $C^\infty$) function at a point of a manifold always extend to a global smooth function?

Obviously this doesn't hold if we replace "smooth" with something like "analytic" or "regular," which are the contexts I'm more familiar with. And obviously we can't extend a smooth function defined ...
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Is $X$ diffeomorphic to its diagonal?

I missed geometry in school, I'm trying to fill in the gaps by reading Pollack's Differential Topology. Am I doing this right? This is #16 in the first section. Show the diagonal $\Delta$ of $X\times ...
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Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
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Isotopy between two open disks on a surface

So I have a (compact) surface $\Sigma$ and two open disks on the surface call them $A$ and $A'$ such that the intersection contains a simple curve $P$. What I want to do is construct an isotopy ...
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derivative along a curve with respect to a given vector field

This is question 7 of Do carmo's differenital geometry of curves of surfaces 3.4. Define the derivative $w(f)$ of a differentiable function $f:U \subset S\rightarrow R$ relative to a vector field $w$ ...
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68 views

Diffeomorphism on $\mathbb C$

Let $P= a_{0} z^{n} +a_{1} z^{n-1}+ \cdots +a_{n}$, with $ a_{0} \neq 0,z \in \mathbb C$. I don't know why $P$ fails to be a local diffeomorphism only at the zeros of the derivative polynomial ...
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What is nonhomogeneous linear mapping?

In Milnor's Topology from the differentiable viewpoint, page 3, he said: One thinks of the nonhomogeneous linear mapping from the tangent hyperplane at $x$ to the tangent hyperplane at $y$ which ...
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117 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
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Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or ...
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(Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I ...
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Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
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Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
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If a smooth manifold X is covered by an odd sphere, then X is orientable.

In solving some old qualifying exam questions, I've been thoroughly stumped. If a smooth manifold $X$ is covered by an odd dimensional sphere, then $X$ is orientable. I see this question has ...
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Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...
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Smoothness does not depend on the choice of atlases

Here is a part of a lecture note: I need some help to solve the exercise. I want to show that if $\psi\circ f\circ\phi^{-1}$ is differentiable and $\alpha, \psi$ and $\beta,\phi$ are in the same ...
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558 views

Understanding the definition and meaning of cotangent space

I would like to understand definition and also meaning of cotangent space. Let's first of all define tangent space: generally we know that equation of tangent line of function $f(x)$ at point $x_0$ ...
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Signature form of $S^2 \times S^2$

Let $M=S^2 \times S^2$ be the product of two copies of the $2$-sphere. We have that $dim(M)=4$. So we can define the intersection form $$ I_{S^2 \times S^2} := H^2(M, \mathbb{Z}) \times H^2(M, ...
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Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
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Show: $W:=\left\{(x_1,x_2,x_3)\in\mathbb{R}^3:-1<x_i<1, i=1,2,3\right\}$ is a 3-dim. submanifold of $\mathbb{R}^3$

Use two different argumentations to show that $$ W:=\left\{(x_1,x_2,x_3)\in\mathbb{R}^3:-1<x_i<1, i=1,2,3\right\} $$ is a 3-dim. submanifold of $\mathbb{R}^3$. 1) ...
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Restriction of smooth functions.

Consider the following question: Suppose that $X$ is a subset of $\mathbb{R}^n$ and $Z$ is a subset of $X$. Show that the restriction to $Z$ of any smooth map on $X$ is a smooth map on $Z$. (Note: A ...
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Every point in a k-manifold has a neighborhood diffeomorphic to $\Bbb{R}^k$

The problem comes from Alan Pollack's Differential Topology, pg. 5. Suppose that X is a k-dimensional manifold. Show that every point in X has a neighborhood diffeomorphic to all of $\Bbb{R}^k$. I ...
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Natural diffeomorphism between $T\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{S}^n\times\mathbb{R}^{n+1}$

I need to show that there is such a diffeomorphism between these spaces. I've tried looking at the 'faces' of elements on both spaces. It went like this: every element in $T\mathbb{S}^n\times ...
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Turning higher spheres inside out

I know that one can turn a sphere $S^2$ in $\mathbb{R}^3$ inside-out having at each time an immersion of $S^2$ into $\mathbb{R}^3$. It is called Smale paradox. There is beautiful animation about that. ...
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Is the closure of an open bounded convex set already a ball?

Does the "closure of an open bounded convex set in ${R}^n$ symmetric wrt. the origin" has to be already homeomorphic to a ball? (My motivation is this: one version of Borsuks theorem says that if ...
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show a map of complex projective space is lefschetz

This is a problem from a qualifying exam. Let $A \in GL_{n+1}(\mathbb{C})$. Then $A$ defines a smooth map on $\mathbb{CP}^n$ by $A \cdot [z] = [Az]$ for $[z] \in \mathbb{CP}^n$. We will denote this ...
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Find the critical points of this function

Let $M={(x,y,z,w)∈R^4|x^4+y^4 + z^2 + w^2 = 1}$and let $f:M \rightarrow R$ be given by$f(x,y,z,w)=x^3 - z.$ a) Show that M is a manifold. b) Find the critical points of f. Part a is easy but how do ...
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Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
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Is there a name for this particular class of topological space?

This is a simple question, but I can't figure out the name for this class of topological space. Say you start with the affine space $\mathbb{R}^n$ for finite n, and equip it with a metric. Now, say ...
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Is the number 8 special in turning a sphere inside out?

So after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here? ...
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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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4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
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464 views

Constant Rank Theorem and Submanifolds

I'm related to my previous question here. The problem is: I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that ...
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Is this enough to show that this map has constant rank?

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is ...
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On the proof that the inverse value set of a regular value is a submanifold

I have a doubt on the proof of the following, well-known theorem: Let $f:M^m\rightarrow N^n$ ($m\geq n$) be a $C$ map, $r\geq 1$.If $y\in f(M)$ is a regular value, then $f^{-1}(y)$ is a $C$ ...
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Typo or additional hypotheses needed in problem 5, p.71 of Bredon's Topology and Geometry?

The problen can be foun in p.71 of Topology and Geometry. I state it below for convenience. Problem: Consider the half open real line $X=[0,\infty)$. Define a functional structure $F_{1}$ by taking ...
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Isotopy preserving inverse image $f_t^{-1}(V)$ of a homotopy

During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it... I'd be ...
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Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
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Immersion is a diffeomorphism.

Clearly I suppose to put the condition $n > 1$ in use. So my proof must went wrong.. Could someone help me take a look at it? Thanks! Suppose $X$ is a smooth, compact, connected $n$-manifold ...
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What is the killing field on $S^2$ and $SO[3]$?

What is the killing field on $S^2$ and $SO[3]$? I understand the structure on $S^1$, but not sure about how the vectors work on $S^2$. Thanks in advance!
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A functional structure on the graph of the absolute value function

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the graph of the absolute value function. That is, $X=\{(x,|x|) : x\in\mathbb{R})\}$. We define a functional structure on $X$ by restricting ...
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$G$ must hit the origin. - Is this proof legit?

The map $F : \mathbb{R}^3 \to \mathbb{R}^3$ given by $F(x, y, z)=(-z, -x, -y)$ restricts to a map $f : S^2 \to S^2$ from the 2-sphere to itself. Show that if $G$ is another map of Euclidean ...
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Show that the Euler characteristic of $O[3]$ is zero.

Show that the Euler characteristic of $O[3]$ is zero. Consider a non zero vector $v$ at the tangent space of identity matrix. Denote the corresponding matrix multiplication by $\phi_A$. Define ...
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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 ...