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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Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...
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112 views

If a connection on a principal $G$-bundle restricts to an $H$-subbundle, must its holonomy lie in $H$?

Let $P \to M$ be a principal $G$-bundle, equipped with a principal connection $D$. Let $Q \subset P$ be a principal subbundle with fiber $H$, where $H \leq G$ is a (let's say closed and connected) ...
6
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2answers
199 views

Why 'closed differential forms' are called 'closed'?

As is well known a differential form $ \omega $ is called closed differential form if it satisfies $ \mbox{d} \omega = 0 \, \, (\ast) \, $ where $ \mbox {d} $ is the exterior derivative. I think the ...
5
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0answers
126 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
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97 views

The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
9
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1answer
148 views

What are necessary and sufficient conditions for the product of spheres to be paralellizable?

Okay, so I found the result that the tangent-bundle of any product of spheres is parallizable, given that some element of the product is either $S^1$, $S^3$, or $S^7$. I prove this as follows, first ...
2
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0answers
56 views

uniqueness of Hopf invariant

(Hopf invariant, page 427 of A. Hatcher's Algebrac Topology): Let $f: S^m\longrightarrow S^n$ with $m\geq n$. We can form a CW-complex $C_f$ by attaching a cell $e^{m+1}$ to $S^n$ via $f$. The ...
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1answer
77 views

Computing the cohomology $H^1(T,\mathbb{Z})$

Let $T$ be a maximal torus of a compact Lie Group. Then how can we compute the first cohomology $H^1(T,\mathbb{Z})$?
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1answer
44 views

Dependence on the class of differentiability between manifolds and maps

Maybe a silly question, but in some books (like "Differential Geometry - Manifolds, Curves, Surfaces - Gostiaux and Berger"), when differentiable maps of class $C^s$ are defined, we have something ...
2
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0answers
62 views

$\mathbb{RP}^2$ does not embed into $\mathbb{R}^3$: reduction to the differentiable case

It is not difficult to see that the real projective plane cannot be embedded into $\mathbb{R}^3$ as a differentiable submanifold (for example one can easily show that the complement would consist of ...
1
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1answer
32 views

Involutive Properties of Space-structures on Smooth Manifolds

I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, ...
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1answer
199 views

Understanding the Concept of Monodromy; case of Lefschetz Fibrations.

My question is on the concept of monodromy around critical points in a Lefschetz fibration $p: M^4 \rightarrow S^2$ (and monodromy in general), where $M^4$ is a 4-manifold and $S^2$ is the 2-sphere. ...
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0answers
67 views

Prove that a canonical bundle is trivial

Consider a function $f \in C^{\infty}(\mathbb{R}^n)$, $y \in Reg(f), M=f^{-1}(y)$. Prove that the canonical bundle of M is trivial. I have an hint but I don't know how to use it: consider the open ...
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1answer
74 views

Definition of “Representing” a Handlebody (Lefschetz Fibration)?

Sorry, I could not find a clear explanation of the meaning of the word represented in the following:"any 4-dimensional 2-handlebody W can be represented by a topological (achiral) Lefschetz fibration ...
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1answer
96 views

On the “regularity” of the boundary of an open set

Let $M = \mathbb{R}^2$ (or more generally, let $M$ be a topological manifold) and let $\Omega$ be an open set in $M$. I'm considering the following regularity conditions for the boundary of $\Omega$: ...
5
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1answer
88 views

Second Stiefel-Whitney Class of a Five Manifold

There is a unique rank 4 nontrivial orientable vector bundle over the 2-torus, denote this by $p:E\rightarrow T^2$. Denote the associated sphere bundle by $S(E)$. Then since $S(E)$ is orientable, the ...
0
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1answer
78 views

On the integration of differential forms

In my notes the integration of a differential form on an oriented manifold $M$, with $\{(U_\alpha,\phi_\alpha) \}$ oriented atlas, is defined as: $\int_M \omega = \sum_{i \in \mathbb{N}}\int_M ...
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1answer
47 views

Proving that charts are related

Let $A$ be an atlas on the set $M$ and let $ x: U \to x(U) $ and $ y : V \to y(V )$ be bijections from subsets $U, V \subset M$ to open sets $x(U), y(V ) \subset \mathbb{R^n} $. Show that if the ...
1
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1answer
125 views

prove that this function is an immersion

How I can show that $F: \mathbb{R} \rightarrow \mathbb{R}^2$ defined by $F(t)=(\cos(t),\sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p\in\mathbb{R}$, $dF_p$ is injective. ...
3
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1answer
179 views

prove that a function is an immersion

How I can show that $F \colon \mathbb{R} \to \mathbb{R}^2$ defined by $F(t)= (\cos (t), \sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p$,$dF_p$ is injective. I have ...
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1answer
103 views

Product manifolds

I have a question on the product of two manifolds. I have $M, N$ two real manifolds (with a smooth differentiable structure), with $\partial M=0$. I have showed that $M\times N$ has a natural induced ...
2
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2answers
74 views

homeomorphism between maninifolds

Exist a local homeomorphism between the manifolds with boundary $[0,1) \times [0,1) $ and $\mathbb{R}^{2}_{+}$? I don't think that a local homeomorphism like this can exist..
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1answer
34 views

Determine $n$ so that manifold is locally homeomorphic to $\mathbb{R}^n$?

I just started studying smooth manifolds. The definition of a topological manifold requires a topological space to be locally Euclidean: homeomorphic to $\mathbb{R}^n$. I know some examples, like how ...
2
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1answer
52 views

integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
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1answer
68 views

What is the best (in terms of effectively building understanding) direction from which to approach manifolds?

Thedore Frankel's book The Geometry of Physics presents Manifolds right away in Chapter 1 in the following manner: Introduce the Euclidean space $\mathbb{R}^N$ only as "the most important manifold". ...
2
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1answer
96 views

Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...
2
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1answer
58 views

$S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$

$G_2(\mathbb{R}^4)$ is the Grassmannian manifold of two-dimensional subspaces of $\mathbb{R}^4$. I would like a detailed proof. Can it be done explicitly? I mean, showing the map and checking its ...
2
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1answer
50 views

Is there a topological space and meanwhile a linear space such that its vector addition is discontinuous but scalar multiplication is continuous?

The title is the question. Does there exist a topological space and meanwhile a linear space X such that its vector addition operation is discontinuous but scalar multiplication operation is ...
2
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1answer
71 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
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0answers
41 views

Compute $\int_0^{x_0}f^\prime(x)$ and $\int_{x_0}^{x_1}f^\prime(x)$

Suppose $f: S^1 \to S^1$, and $f(x_i) = y$, where $x_i$s are the preimage of a regular value $y$. Then how can I compute $\int_0^{x_0}f^\prime(x), \int_{x_0}^{x_1}f^\prime(x),$? I realize that ...
0
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1answer
145 views

Condition for Differential Forms to Pass to the Quotient

everyone: I was reading this question : What do we mean when we say a differential form "descends to the quotient"? which is related to mine. But the reply given did not answer my question ...
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35 views

The transverse root for a vector field

I encountered the term, so may I ask - what is the transverse root for a vector field? Thank you~
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47 views

The piecewise-smooth homotopy

Let $M$ be a smooth manifold. If $\gamma_1:I \rightarrow M$ and $\gamma_2:I\rightarrow M$ are two piecewise-smooth homotopic curves (rel to endpoints), then can we find a map $f:I\times I\rightarrow ...
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1answer
77 views

A homework problem about Tangent space of Lie groups at the identity element

This is one of our homework problem. Let $G$ be a Lie group be defined as a manifold with group structure such that the map $F:G\times G \mapsto G, F(a, b)=ab^{-1}$ is smooth. Show that ...
3
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1answer
372 views

Section of a vector bundle as a submanifold

I am currently working on one part of a problem surrounding sections and submanifolds. Given a real vector bundle $\pi: E\rightarrow M$ of rank k, with a smooth global section $s:M\rightarrow E$, can ...
3
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0answers
58 views

Crosscap function in $\mathbb{R}^4$ - and how to show it is proper?

I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$ My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue): Q1: Is ...
2
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0answers
105 views

Is this curve a minimal geodesic?

Please help me identify this curve. I strongly feel that it is a minimal geodesic, but I cannot show it. Suppose that $M$ is a Hadamard manifold (i.e., complete, simply-connected smooth Riemannian ...
1
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1answer
114 views

Diffeomorphism $S^1\to S^1$ extends to a diffeomorphism $D^2\to D^2$

Suppose $\phi:S^1\to S^1$ is a diffeomorphism. We can think of $D^2$ as the half sphere: Actually this half sphere is just the rotation of the semicircle $C$ from $0$ to $\pi$ (I mean we rotate ...
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0answers
84 views

transverse homotopy

Let $f,g:M \rightarrow N$ two smooth maps between smooth manifolds that are smoothly homotopic by $F$. Suppose also that $f$ and $g$ are transverse to a submanifold $A$ of $N$. I know that transverse ...
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1answer
264 views

a Problem about Degree of Map between Spheres

When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
2
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1answer
242 views

Are there degree-1 maps from $S^2 \times S^3 \rightarrow S^5$ or from $S^5 \rightarrow S^2\times S^3$?

This is a question from a past qualifying exam I am stuck on: For a smooth map $f:M\rightarrow N$ between smooth, compact, oriented $n$-manifolds, the degree of $f$ is the unique integer $k$ such ...
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1answer
62 views

$GL(n,\mathbb{C})$ is semi-locally simply connected.

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.
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Loops in $RP^2$

We know that $\pi_1(RP^2)=Z_2.$ How do non-trivial loops in $RP^2$ look like? (If $RP^2$ is the upper hemisphere $D$ with antipodal points of $\partial D$ identified)
2
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1answer
59 views

Intersection number on $S^k$

Is the intersection number on $S^k$ is always zero? Choose $X$ a compact submanifold of $S^k$, and $Z$ a closed submanifold of complementary dimension, viewing $I_2(X,Z) = I_2(i, Z)$ where $i: X ...
3
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0answers
161 views

Moebius strip as a fibre bundle

I've alrealdy asked this question, but now I have more clear ideas, so I'm going to ask again and see if I'll understand a bit more. It's about the trivialisation of the Moebius strip as a bundle on ...
2
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0answers
31 views

Submersions and complex structure

Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ ...
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1answer
40 views

Show $\exists p \notin f(X) \cup Z$ by Sard

Guillemin & Pallack P83, Ex 2.4.9: Suppose $X$ compact with $0< \dim(X) < k$ and $f: X \to S^k$. Suppose $Z \subset S^k$ a closed submanifold with $\dim(X) + \dim(Z) = k.$ Show that ...
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1answer
34 views

$dg_y$ carries a subspace of $T_y(Y)$ onto $\mathbb{R}^l$

The question arises from Guillemin and Pallack Page 28 above the frame: $dg_y$ carries a subspace of $T_y(Y)$ onto $\mathbb{R}^l$ precisely if that subspace and $T_y(Z)$ span all of $T_y(Y)$. I ...
2
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1answer
119 views

Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
6
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3answers
252 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...