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Questions tagged [differential-topology]

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.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Existence of a "special atlas"?

Let $M$ be a smooth manifold and $\mathcal{A} =\{(\varphi_j, U_j) \}_{j\in J}$ an atlas of $M$. I know that exists a refinement $\mathcal{V}$ of the cover $\{U_j\}_{j\in J}$, such that $\mathcal{V}$ is locally finite. Now I would like to demonstrate…
Matheus Manzatto
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Are diffeomorphisms dense in $C^0(M)$ for a compact manifold M?

Given a homeomorphism $f$ on a compact manifold $M$, can you find a $C^1$ diffeomorphism within an arbitrarily small neighborhood of $f$? In other words, can you make an arbitrarily small perturbation to smooth out $f$? It seems like a natural…
P. May
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Question on the proof of the Stability Theorem in Guillemin & Pollack

I'm having trouble with the following part in the proof of the stability theorem on Pg 36 of Guillemin and Pollack's Differential Topology: They write that since $X$ is compact, it follows that any open neighbourhood of $X\times \lbrace 0 \rbrace$…
Apoha
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Show that an embedding from a compact, smooth manifold to a connected smooth manifold is a diffeomorphism.

Let $M,N$ be nonempty, smooth manifold of the same dimension. Let $N$ be compact and $M$ be connected. Suppose that $f:N\rightarrow M$ is an embedding. We want to show that $f$ is a diffeomorphism. Since $f$ is an embedding, then we know that…
user394412
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Symmetry of the linking number

(Problem 13 of Milnor's Topology from the differentiable viewpoint). Let $M^n,N^m\subset \mathbb{R}^{m+n+1}$ be compact, oriented differential manifolds without boundary of dimension $n$ and $m$. Consider $\phi:M^n \times N^m\rightarrow…
PP123
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Is $SO(n)$ diffeomorphic to $SO(n-1) \times S^{n-1}$

There is a fibration $SO(n-1) \mapsto SO(n) \mapsto S^{n-1}$, from basically taking the first column of the matrix in $\mathbb{R}^n$. Is this fibration trivializable?
vukov
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Why does this limit define a linear map between the tangent spaces?

Let $U \subset \mathbb{R}^n$, $V \subset \mathbb{R}^k$ be open, $f\colon U \to V$ a smooth map, $x \in U$. For $h \in \mathbb{R}^n$, let $$ d_x(h) := \lim_{t \to 0} \frac{f(x+ th) - f(x)}{t} $$ Now, in "Topology from the differential viewpoint",…
Vincent
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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 $TM_{x},TN_{x}$ together with that of $X$. Define $[\partial N,…
Bombyx mori
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what is the difference between a differential structure and a maximal atlas?

I am afraid I am missing some subtlety here. they both have pages on wiki: differential structure: https://en.wikipedia.org/wiki/Differential_structure atlas: https://en.wikipedia.org/wiki/Atlas_(topology)#Charts
Alg
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Discussion of Poincaré-Bendixson Theorem

Poincaré-Bendixson Theorem states that let $\mathbf{F} : \mathbb R^2 \to \mathbb R^2$ be a $C^1$ vector field in $\mathbb R^2$ and consider the system $\mathbf{x'} = \mathbf{F(x)}$. Suppose $K$ is a set in $\mathbb R^2$ such that: $(1)K$ is closed…
Martin
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How to show that $\Bbb R^m$ is not diffeomorphic to $\Bbb R^n$ when $n \neq m$

Suppose that there is indeed an diffeomorphism $f: \Bbb R^m \to \Bbb R^n$, and $f(a) = 0$. Then $f \circ f^{-1} = identity$. In class my teacher asserts that $Df^{-1}_a \cdot Df_a = identity$ and $Df_a \cdot Df_a^{-1} = identity$. Why this is true…
Keith
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Independent of Choice of Local Coordinates/Global Tensor Field

Let $M$ be a smooth manifold, and let $x_ 1 , . . . , x_n$ be a local coordinate system defined on an open set $U ⊆ M$. Consider the $(1, 1)$-tensor field $C$ defined on $U$ in local coordinates by $C…
Laura
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Pullbacks of volume forms on the circle fail to extend to closed forms

I am working through this problem: Let $M$ be a compact oriented 3-manifold with boundary, where the boundary is $\partial M=S^1\times S^1$. Let $\theta_i\in\Omega^1(\partial M)$, $i = 1, 2$ be the 1-forms obtained by pulling back the standard…
Android Netizen
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How to calculate the degree of this Gauss map?

In reviewing the familiar Poincare-Hopf theorem I come across the following question: Suppose $x$ an isolated 0 of $V$. Pick up a disk around $x$ in its neighborhood. Calculate the degree of the map $$u:\partial D \rightarrow…
Bombyx mori
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A Couple of Normal Bundle Questions

We are working through old qualifying exams to study. There were two questions concerning normal bundles that have stumped us: $1$. Let $f:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}$ be smooth and have $0$ as a regular value. Let…
J126
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