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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195 views

$(n - 1)$-dimensional submanifold of the manifold $\mathbb R^n$

Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$. Let $0 \neq b \in \mathbb{R}$. Show that the surface $M = \{x\in \mathbb{R}^n \mid x^T A x = b\}$ is an $(n - 1)$-dimensional ...
22
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2answers
745 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
2
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0answers
86 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
5
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1answer
139 views

Proving that $O(n,m)$ is simply connected.

My question is the following: Under which conditions on given integers $n\le m$ is $$O(n,m) = \{A \in \mathbb R^{m\times n} : A^TA = \mathbf 1\}$$ simply connected? Does anyone know a reference for ...
0
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1answer
287 views

Co-homology Groups of the Torus

I wanted to explain to me, or give me a reference of how to calculate the cohomology groups of the complex and real, torus $\mathbb{T}^2$. I want to use this as an example in a seminar that I will ...
1
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1answer
136 views

How to construct a map from $\mathbb{ S}^2=\{(a_1,a_2,a_3) | a_1^2+a_2^2+a_3^2=1\}$ to $\mathbb{ RP}^2$?

How would I construct the map? Once constucted, would I be right in saying that there is no Diffeomorphism to map back? As in $\mathbb{RP}^2$ a closed curve would have to have either $2$ points that ...
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3answers
401 views

Infinitely sheeted covering spaces!

I was wondering what the fundamental group of an infinitely sheeted covering space of say some surface might be? I'm thinking it should be an infinite cyclic group, but this is more intuitively, i ...
4
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1answer
242 views

Morse index and Euler characteristic

I found the following problem and I couldn't solve it. Let $X$ be a compact manifold and $f$ a Morse function (all of its critical points are non degenerate) on $X$. Prove that the sum of the Morse ...
8
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1answer
285 views

Existence of geodesic on a compact Riemannian manifold

I have a question about the existence of geodesics on a compact Riemannian manifold $M$. Is there an elementary way to prove that in each nontrivial free homotopy class of loops, there is a closed ...
1
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1answer
75 views

2-Manifold an image of the unit disc?

Is every 2 dimensional manifold whose boundary is a cycle, a continuous image of the unit disc? Maybe it happens if the space is good enough? I wanted to prove an equality between two definitions I've ...
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46 views

Linking integral unchanged over continuous deformations

Say we first have two curves, $C_1$ and $C_2$ which are knotted together. Let $C_2'$ be a continuous deformation of $C_2$ such that $C_2$ does not cross $C_1$ as it is deformed into $C_2'$. How ...
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0answers
76 views

How to prove this isotopy exist?

Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that ...
20
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2answers
1k views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
3
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1answer
59 views

Are these functions homotopic?

Let $\gamma$ be a smooth, simple, closed curve and let $f : \gamma \to S^1$ assign to each $x \in \gamma$ the unit normal vector there. We can find a diffeomorphism $g: \gamma \to S^1$ and define the ...
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3answers
152 views

A manifold being orientable vs oriented.

I read that an oriented manifold is a manifold with a choice of orientations for each tangent space so that for $p \in M$, there is an open set $U$ and a collection of vector fields $X_1,...,X_n$ so ...
5
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1answer
217 views

Examples of special sphere bundles

I'm interested in examples of sphere bundles which do not arise from vector bundles. I'm not quite clear about the following. So please let me know if anything is false. I believe that a ...
3
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1answer
498 views

Splitting of the tangent bundle of a vector bundle

Let $\pi:E\to M$ be a rank $k$ vector bundle over the (compact) manifold $M$ and let $i:M\hookrightarrow E$ denote the zero section. I'm interested in a splitting of $i^*(TE)$, the restriction of the ...
2
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1answer
92 views

Question about differential of embedding

For any $C^{\infty}$ manifold $M$, the tangent bundle $TM$ of $M$ is also a $C^{\infty}$ manifold. Hence we can think about the differential $df:TM\rightarrow TN$ of maps $f:M\rightarrow N$ between ...
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1answer
101 views

Regular value of $g \circ f$ is a regular value of $g$

Given smooth maps $f: X \to Y, g: Y \to Z$, where $X, Y, Z$ are boundaryless, compact manifolds of dimension $n$, is the statement in the title true?
0
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1answer
268 views

De Rham cohomology question

I'm trying to compute a certain DeRham cohomology. Consider $M = S^n-C$, where $C$ is the disjoint union of closed disks $C = \cup_{i=1}^m D_i$, and $m,n \geq 1$. How can we compute the cohomology ...
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1answer
153 views

A “Manifold with Boundary” Question

I'm given the following scenario: Letting $U$ be an open subset of $\mathbb{R}^n$ and $f,g:U\rightarrow \mathbb{R}$, two smooth functions such that $f(\vec x)\lt g(\vec x)$ for all $\vec x\in U$, ...
3
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1answer
168 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
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1answer
158 views

Diffeomorphism between a triangle and a square?

Is there always a diffeomorphism between $(0,1)^2$ and any given (not degenerate) triangle?
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1answer
634 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
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0answers
34 views

Other definitions of singularity

Many definitions of a singularity of a manifold $X^n$ are concentrating on the defining equations of it and the vanishing of the (partial) derivatives. My questions: What if $X^n$ isn't algebraic ...
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2answers
300 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
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291 views

How to prove a manifold is diffeomorphic to Euclidean space?

Problem is this: suppose a manifold $$M=\bigcup_{n\in\mathbb{N}} U_n,$$ where each $U_n$ is diffeomorphic to Euclidean space, and $U_n$ is contained in $U_{n+1}$. Then please show that $M$ is ...
3
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1answer
104 views

Suspension Operation on the Pontryagin-Thom Construction

I have a feeling that this is well-known: View the Pontryagin-Thom construction as the bijective correspondence between $[M,S^r]$ and the set of (appropriate equivalence classes of) framed ...
2
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3answers
158 views

Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
3
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2answers
434 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads: Use Fubini´s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
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348 views

is the fixed set of a smooth involution a submanifold?

Let $f:X\rightarrow X$ be a smooth map of a smooth manifold with $f^2=\operatorname{id}$. Is the subset $\{x\in X\mid f(x)=x\}$ a smooth submanifold? I tried to find an argument with the implicit ...
2
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1answer
304 views

Trivial bundle on sphere

I have an exercises as follows: Let $E$ be a trivial bundle on $S^n$. Prove that the Whitney sum $TS^n\oplus E$ is also trivial. The hint is using the normal bundle of $TS^n$, but I don't know how to ...
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2answers
148 views

Are these charts on the circle compatibly oriented?

I've tried a few methods but I can't seem to work this one out. Consider the charts $$f(s) = (\cos s, \sin s) \in \mathbb{R}^2$$ for $-\pi < s < \pi$ and $$g(t)=(\frac{2t}{t^2 + 1}, \frac{t^2 ...
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2answers
219 views

How to obtain a Morse function on a submanifold of Euclidean space

Consider a smooth $n$-dimensional submanifold $A$ in $\mathbb{R}^{n+1} \times \mathbb{R}$ and the projection $f:\mathbb{R}^{n+1} \times \mathbb{R}\rightarrow \mathbb{R}$ onto the second factor. Is it ...
8
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1answer
143 views

Morse homology of $P^2$

I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is ...
4
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1answer
218 views

Example of symplectic manifold

I wonder why tangent bundle is not symplectic. As you know, cotangent bundle is symplectic. (1) question 1 : Is cotangent bundle isometric to tangent bundle ? ...
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0answers
156 views

Framed cobordism

I have an exercise as follows: "Let $M\subset \mathbb{R}^k$ be a smooth, connected, oriented, compact manifold without boundary of dimension $p$. Let $\Omega^{Fr}_n(M)$ be the set of all equivalence ...
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0answers
26 views

h-cobordism theorem in three category

In wikipedia page, it says that h-cobordism theorem is true for three category=DIFF,TOP,PL. I learned a proof of h-cobordism theorem using Morse function, but it works only in DIFF category. How is ...
9
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3answers
458 views

Why do people care about principal bundles?

I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought ...
2
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1answer
69 views

Is h-cobordism theorem true on smooth category?

On h-coobrdism page of Wikipedia, it says that h-cobordism theory is true for smooth category. As far as I know, it would imply that smooth Poincare conjecture is true for dimension greater than 6, ...
3
votes
1answer
71 views

a question about germs of functions

Let $M$ be a smooth ( real) manifold, if $ p\in M$ and $f\in C^{\infty}(U)$ ($U$ is an open subset of $M$), the symbol $[ f]_p$ indicates the smooth germ of $f$ at $p$ . Consider the following set ...
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1answer
63 views

Transversality of a mapping

The question I have is: Show that the mapping $g:R^2 \rightarrow R^3 $ given by : $y_1 = x_1 (x_1 ^2 -x_2 ^2 +1), y_2=x_2, y_3=x_1 ^2$ is transversal to all lines $y_2 = \textrm{constant}$ in the ...
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1answer
227 views

Is the tangent bundle the DISJOINT union of tangent spaces?

Let $M$ be a smooth manifold and consider the Lee's definition of the tangent space $T_pM$ (so $T_pM$ is the vector space of derivations at $p$). The canonical definition of tangent bundle (as set) of ...
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1answer
159 views

The connection of Morse function

Suppose $M$ and $N$ are two manifolds, $f$ is a Morse function on $M$, $g$ is a Morse function on $N$, can you find a new manifold $P$ as the connection of $M$ and $N$ and a Morse function $h$ on the ...
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1answer
128 views

How to conclude that a path is non-trivial element of $\pi_1(M)$

Let $M^3$ be a compact manifold. If $\mathbb{RP}^2$ is embedded in $M$. Suppose, by contradiction, that $i_\sharp: \pi_1(\mathbb{RP}^2) \longrightarrow \pi_1(M)$ is non-injective and that the normal ...
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0answers
75 views

show that subset of complex projective space is a submanifold

Let n, m $\in \mathbb{N}$. I'm trying to show that $M(n,m) = \{[z_0 : z_1 : … : z_n] \in \mathbb{C}P^n | \sum^{n}_{i=0} z^m_i = 0\}$ is a submanifold and codim(M(n,m))=2. My idea was to use ...
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2answers
265 views

Is there a compact, connected manifold $M$ with boundary $S^2$ not diffeomorphic to $D^3$?

Is there a compact, connected, smooth 3-manifold $M$ with boundary $S^2$ not diffeomorphic to $D^3$ (the closed unit ball)? If so, what is it? The compactness condition rules out the complement ...
2
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1answer
258 views

How show that a surface embedded is non-orientable?

Let be $M$ a compact $3$-manifold. If $\Sigma$ is a embedded surface in $M$, such that $\Sigma$ is homeomorphic to $\mathbb{RP}^2$. If $i: \pi_1(\Sigma) \longrightarrow \pi_1(M)$ is not injective, ...
0
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1answer
120 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
3
votes
2answers
385 views

Is the Empty set an orientable manifold?

The empty set can be regarded as an object in the category of smooth manifolds, at least for technical considerations. Is the empty set an orientable manifold?