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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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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501 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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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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194 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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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 ...
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1answer
96 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 ...
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133 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.
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346 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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252 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 ...
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1answer
210 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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147 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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147 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
126 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 ...
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121 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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139 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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25 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 ...
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3answers
319 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 ...
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1answer
67 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, ...
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58 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
47 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
148 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
126 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
107 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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66 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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180 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 ...
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1answer
206 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, ...
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1answer
107 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 ...
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2answers
263 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?
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100 views

Identification of integration on smooth chains with ordinary integration

Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
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$D^m\cup_h D^m$, joining $D^m \amalg D^m$ along the boundary $\partial D^m$

Given an orientation-preserving diffeomorphism $h: \partial D^m \to \partial D^m$, we can glue two copies of the closed unit disk $D^m$ along the boundary by identifying $x \sim h(x)$ to form the ...
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170 views

Fundamental Group!

Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
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118 views

How does smoothness prevent “singularities”?

This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. ...
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45 views

Embed connected sum of submanifolds in the ambient manifold

The book Differential Manifolds by Kosinski claimed we can embed the connected sum of two submanifolds of codimension $>1$ into the ambient manifold. I am wondering why we need the extra condition ...
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1answer
72 views

$Gr_2^+(\mathbb R^4) \cong S^2 \times S^2$

Let $Gr_2^+(\mathbb R^4)$ be the oriented Grassmanian of 2-planes in $\mathbb R^4$. How would one go about showing that this is diffeomorphic to $S^2 \times S^2$?
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50 views

Topologies in the space $C^r(M,N)$

I am currently taking a course in Differential Topology where the book we are using the one by Hirsch. I have had some troubles trying to prove that the weak topology makes the composition of maps, ...
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1answer
160 views

Reference for topology and fiber bundle

I am looking for an introductory book that explains the relations of topology and bundles. I know a basic topology and algebraic topology. But I don't know much about bundles. I want a book that ...
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1answer
553 views

Line with a knot in $\mathbb R^3$ isotopic to standard embedding $\mathbb R\subset \mathbb R^3$?

I have come across the following exercise in Kosinski's 'Differential Manifolds': Exercise: Consider an imbedding $\mathbb R\to \mathbb R^3$ where the image is "the line with a knot": ...
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147 views

Do we really need embedding for this?

Question 8-8 in Lee's Introduction to smooth manifolds asks us to show that if $M \subset N$ is an embedded submanifold then it is closed iff the inclusion map is proper. Equivalently, a smooth ...
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1answer
81 views

Need help understanding proof about critical values of the determinant map.

In , problem 5 the author shows that the differential of the determinant map $d(\det)_A$ for an invertible matrix $A$ is nonsingular by only showing that $d(\det)_A(A) \ne 0$. I don't really see why ...
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Uniform Poincaré-Wirtinger constant for diffeomorphic domains?

Consider a bounded and connected open set (with regular boundary) $\Omega\subset\mathbb{R}^d$ and a family of $\mathscr{C}^1$-diffeomorphisms $(\varphi_t)_{t\in[0,\alpha]}$ all in ...
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2answers
153 views

Question about definition of a smooth manifolds (do all transition maps have to be smooth)?

I need to clear up my confusion on the definition of a smooth manifold. So we say that $M$ is a smooth manifold (of dimension $n$), if $M$ is Hausdorff and if every $x \in M$ is contained in a ...
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4answers
206 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
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53 views

How to prove $[\partial M, N]=[\partial N, M]$?

Consider $M,N$ submanifolds of some manifold $X$ such that $\dim M+\dim N=\dim X$. For $x\in M\cap N$, let $\langle M_{x},N_{x}\rangle$ denote the index by matching local orientation of ...
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1answer
77 views

How to show $\mathbb{R}^{n}+\mathbb{S}^{n}=\mathbb{R}^{n}$?

I want to ask how to show $\mathbb{R}^{n}+\mathbb{S}^{n}\cong \mathbb{R}^{n}$ as connected sums where the isomorphism is a differeomorphism between $\mathbb{R}^{n}$ and $\mathbb{R}^{n}$. The proof in ...
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37 views

Existence of a smooth function with a given kernel

Let $K\subset \mathbb{R}^n$ be a closed set, then is there existing a smooth function $f\in C^{\infty}(\mathbb{R}^n,\mathbb{R})$, such that $$ (1)\quad f\ge 0, $$ $$ (2) \quad f^{-1}(0)=K. $$
4
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1answer
140 views

Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
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1answer
83 views

Gluing axiom of a TQFT

In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT. When they come to prove the gluing axiom, they just mention that "...This statement is ...
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169 views

On the converse of Sard's theorem

Let $f: M \rightarrow N$ be a smooth map between two submanifolds of $\mathbb{R}^{m}$, $\mathbb{R}^{n}$ respectively. Sard's famous theorem asserts that the set of critical values $C$ of $f$ has ...
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1answer
47 views

Possible lengths of geodecics

Let $M$ be a compact manifold (w/o boundary). Suppose that there is no closed geodesics on $M$ of length precisely $C$. I am trying to prove that there is an open cover $\{U_j\}$ of $M$ and $\epsilon ...
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1answer
513 views

Prerequisite for Differential Topology and/or Geometric Topology

What are the prerequisites to learning both or one of the items? Consider that one will have done some of the "core" classes like Differential Geometry, Real Analysis, Abstract Algebra and POint-Set ...