# Tagged Questions

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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### Local bisections of Lie groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
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### If $f\colon X\to Y$ is nullhomotopic, then $I_2(f,Z)=0$ for closed $Z\subset Y$ of complementary dimension?

I think I have a near solution, but one doubt. Problem 2.4.4 on page 83 of Guillemin and Pollack asks If $f\colon X\to Y$ is homotopic to a constant map, show that $I_2(f,Z)=0$ for all ...
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### A(nother ignorant) question on phantom maps

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were ...
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### Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions:

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions: a. There is another point $p_{0}$ such that the ...
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### Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two ...
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### Immersion $\mathbb{S}^n\times\mathbb{R}\to\mathbb{R}^{n+1}$

Immersion $\mathbb{S}^2\times\mathbb{R}\to\mathbb{R}^3$ As $\mathbb{S}^2\times\mathbb{R}$ it's not compact, i can give immersion given by $$(x,y,z,t)\to e^t(x, y, z).$$ or i'm wrong. Could you give ...
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### When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
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### Mapping degree of a diffeomorphism

This might be a bit silly question but I haven't find direct reference. Let $\Omega$ be open, bounded and connected in $\mathbb{R}^n$. Assume that $f:\overline{\Omega}\rightarrow \mathbb{R}^n$ is a ...
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### Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are analytic,...
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### chern class of complex line bundle

Let $\xi:=(\mathbb{C},E,p,B)$ be a complex line bundle, where $B$ is a manifold or CW-complex. How to determine whether the first Chern class $c_1(\xi)=0$ or non-vanishing?
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### Lemma on locally finite open covers

I came across this lemma in Lee's 'Introduction to Smooth Manifolds'. The lemma seems simple enough to prove, but I just can't seem to prove it. It's frustrating me because I know it must be simple. ...
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### Theorem 3.1 from Milnor's Morse Theory

Milnor is in the business of proving that if $f: M \to \mathbb{R}$ is a smooth function, $a < b$, and $f^{-1} ([a,b])$ is a compact subset of $M$ containing no critical points, then $M^a$ is ...
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### Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
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### Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
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### Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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### Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...