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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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 $S^1$ ...
2
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4answers
276 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 ...
2
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0answers
39 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 ...
2
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2answers
54 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
45 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 ...
1
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1answer
33 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 ...
1
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0answers
16 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 ...
2
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1answer
60 views

Is complex projective space simply connected?

I know real projective space isn't simply connected, what about complex projective spaces?
3
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1answer
28 views

Two definitions of framed manifolds/cobordism

One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact ...
1
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1answer
66 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
1
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0answers
56 views

Preimages of smooth maps between manifolds

Suppose $M$ and $N$ are both compact, connected, oriented $m$-manifolds without boundary and $f: M \to N$ is smooth. What additional condition(s) must $f$ satisfy so that there exists at least one ...
1
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0answers
30 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
3
votes
1answer
54 views

Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
1
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0answers
52 views

Almost Every Hyperplane is Transverse to $M$

Let $M$ be an $n$-dimensional manifold embedded in $\mathbb{R}^{n+1}$. I am trying to show that almost every hyperplane in $\mathbb{R}^{n+1}$ is transverse to $M$. To show that I would like to prove ...
5
votes
1answer
57 views

Derivative of the inclusion of a submanifold

I know there are other questions similar to this one, but I just want you to tell me if what I'm doing is rigth and how to improve it. The problem is the following: (I'm using the definitions by ...
2
votes
0answers
25 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
4
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3answers
123 views

Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
8
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1answer
59 views

Parametrizing Walks on Sphere and Torus

This question is very underdeveloped, but I was wondering if there was a map from the sphere to the torus which preserves length of closed curves? I was just thinking about taking a walk on a ...
3
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0answers
66 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
2
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0answers
20 views

Generalizations of the product neighborhood theorem

As far as I know, the Product Neighborhood Theorem for smooth manifolds states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization ...
2
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1answer
56 views

Proving a Certain Smooth Map $S^n\rightarrow S^n$ is a Diffeommorphism

I am given a smooth map $f:S^n\rightarrow S^n$, for $n\geq 2$, whose differential is injective at each point. I am asked to prove that it is a diffeomorphism. Since the differential is injective ...
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2answers
84 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
0
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1answer
58 views

Show That $\dim H_m(\partial M;\mathbb{R})$ is Even

A student asked me this. Suppose that $M$ is a compact, orientable $n$-manifold with boundary. It is a fact that for each $k$ with $0\leq k\leq n$ the vector spaces $H_k(M;\mathbb{R})$ and ...
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0answers
21 views

a question related with morse theory [duplicate]

Show that there exists no smooth function $f:\mathbb{R}^2→\mathbb{R}$,such that $f(x,y)\geq 0$ for any $(x,y)\in\mathbb{R}^2$, with exactly two critical points$(x_1,y_1)\in\mathbb{R}^2$, ...
2
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1answer
44 views

Why is every derivation a vector?

We can see the vectors of the tangent space $T_pM$ to a smooth manifold as velocities of curves. This is elaborated here. Each velocity $\gamma'(0)$ corresponds to a derivation $D_{\gamma}(f) = (f ...
2
votes
2answers
82 views

Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
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0answers
32 views

A question about the existence of a smooth function [duplicate]

Does there exists a smooth function $f: R^2 \rightarrow R$, such that $f(x,y)\ge0$, for any $(x,y) \in R^2$, and $f$ has exactly two critical points $(x_1,y_1), (x_2, y_2) \in R^2$ with ...
0
votes
1answer
38 views

Help understanding a proof in differential geometry

I was reading John Milnor's Topology from the Differentiable Viewpoint and there's a proof of the fundamental theorem of algebra at the end of the first chapter that I don't fully understand. I can ...
0
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0answers
44 views

What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
2
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1answer
85 views

$\mathbb{R}^2$ is a Retract of $\mathbb{R}^5$

Looking through some old qualifying exams, I noticed the question: Let $X\subseteq \mathbb{R}^5$ be homeomorphic to $\mathbb{R}^2$. Prove that $X$ is a retract of $\mathbb{R}^5$. I'm not even sure ...
7
votes
2answers
167 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
3
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0answers
103 views
+100

Avoiding vertical vectors in tangent spaces.

Let $i: [0,1]\hookrightarrow\mathbb{R}^3$ be a smooth embedding. Can I find arbitrary small perturbations $j$ of $i$ s.t. $j(0)=i(0)$, $j(1)=i(1)$ and $j'(t)$ is never a multiple of $(0,0,1)$? More ...
2
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1answer
50 views

Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
3
votes
1answer
80 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
5
votes
4answers
243 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
3
votes
1answer
44 views

Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
2
votes
1answer
53 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
1
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1answer
18 views

natural projection on a slice

I'm currently studying Warner's book "Foundations of Differentiable Manifolds and Lie Groups". Within the proof of the Frobenius Theorem he is constructing a slice $S$ of a coordinate system ...
3
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1answer
25 views

Extensions of diffeomorphisms of $S^2$ and the connectedness of $\text{Diff}^+(S^2)$

In this MO question by Daniel Moskovich, he claims that the fact that every diffeomorphism of $S^2$ extends to a diffeomorphism of $D^3$ implies that $\text{Diff}^+(S^2)$, the group of ...
0
votes
0answers
56 views

Accepted symbol (or way of writing) “A is a subset of B or B is a subset of A”

I am looking for a concise way to write the statement "$A$ is a subset of $B$ or $B$ is a subset of $A$". The context is the Grassmannian and two elements $A,B\in G_k(\mathbb R^n)$ in it. The two ...
7
votes
1answer
114 views

Connected sum in an ambient space

Let $M$ be a smooth c connected $m$-manifold, $N_1$, $N_2$ two smooth disjoint connected $n$-submanifolds with boundary, both contained in the interior of $M$ (if necessary). Assume that $M\setminus ...
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0answers
33 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
3
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1answer
36 views

Does there exist a dipole field on $S^2$ differing by at most a minus sign between antipodal points?

Consider the two-sphere $S^2 \subset \mathbb{R}^3$. By a dipole field on $S^2$, I mean a continuous function $f \colon S^2 \to S^2$ such that (1) $x$ is perpendicular to $f(x)$ for all $x \in S^2$ ...
3
votes
1answer
56 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...
8
votes
1answer
208 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
7
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0answers
74 views

Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
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1answer
30 views

Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
4
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0answers
72 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
2
votes
1answer
44 views

Intuition behind the definition of a derivative by Lang

In Serge Lang's Introduction to Differentiable Manifolds he says that a function $f:U\to F$ is differentiable at a point $x_0\in U$ if there exists a linear map $\lambda$ of $E$ into $F$ such that, if ...
2
votes
3answers
76 views

What is the pushforward of a function (not a vector)

If we have two manifolds $M$, $N$ with the map $f:M \to N$, then this induces a map between their tangent spaces $f_*:T_pM \to T_{f(p)} N$. By duality, another map exists $f^* : T^*_{f(p)}N \to ...