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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Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
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171 views

The degree of every smooth map $\mathbb{R}^n \to \mathbb{R}^n$ is one…

Let $\varphi : M^n \to N^n$ be a proper smooth map between two connected smooth manifolds. Then $\varphi$ induces a linear map $\varphi^* : H_c^n(N) \simeq \mathbb{R} \to H_c^n(M) \simeq \mathbb{R}$ ...
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Extending diffeomorphism to disk

I am trying to prove the following If $f:S^1 \to S^1$ is a diffeomorphism it can be extended to a diffeomorphism $F: D^2 \to D^2$. But I can't seem to prove it. I proved it for homeomorphisms using ...
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98 views

does composition of maps is smooth and one map is smooth imply the other is also smooth?

If $f\circ g$ is smooth and $f$ is smooth, does it follow that $g$ is smooth? Note that I cannot simply take the inverse of $f$. Do I have to use implicit function theorem?
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602 views

Diffeomorphism and determinant of Jacobian

I don't remember where I read it and if I remember it correctly but does the following hold true? If $M,N$ are two (smooth?) surfaces and $f: M \to N$ is a homeomorphism such that $det(J_f)$ (the ...
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Products of homeomorphisms

I was wondering if there is a theorem like "If $f_i:X_i\to Y_i$ are homeomorphisms then $\prod_i f_i : \prod_i X_i \to \prod_i Y_i$ is a homeomorphism" for $I$ finite. What about $I = \mathbb N$? ...
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96 views

Locally Euclidean but not topological manifold

I'm having trouble solving one part of one of the initial exercises of the classic Boothby book "An Introduction to Differentiable Manifolds and Riemannian Geometry" (exercise I.3.1). To be more ...
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509 views

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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68 views

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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240 views

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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21 views

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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33 views

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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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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178 views

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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98 views

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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62 views

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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140 views

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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74 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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208 views

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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179 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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136 views

(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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186 views

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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212 views

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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278 views

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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742 views

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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94 views

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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1k 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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534 views

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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1answer
114 views

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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565 views

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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60 views

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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487 views

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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565 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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158 views

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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683 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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149 views

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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1answer
197 views

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 ...