# 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.

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### Is the restriction map continuous and surjective?

Consider $C^{\infty}(U, \mathbb{R}^{k})$ and $C^{\infty}(V, \mathbb{R}^{k})$, where $V\subset U$, with $U$ and $V$ open subsets of $\mathbb{R}^n$. Is it true that the restriction \begin{align} R:C^{...
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### Can we define differentiation without using a norm?

Since all norms on $\mathbb R^n$ are equivalent, the following question makes sense: Can we define the notion of "differentiability" of a map $\mathbb R^n \to \mathbb R$ without refering to a norm ...
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### Hyperplane intersects submanifold transversally

Let $M\subseteq \mathbb{R}^p$ an n-dimensional submanifold. Show that there is a hyperplane in $\mathbb{R}^p$ that intersects $M$ transversally. My ideas: I shall use the theorem of Sard but I don't ...
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### Simply connected neighborhood of a compact set contained in an open set

To continue this line of thought, say we have a compact, simply connected subset $K$ of a manifold $M$. Can a simply connected neighborhood of $K$ be found? I suppose that if $K$ is a submanifold, ...
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### Does 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?

I want to know Does 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions? My idea is that one can consider 2-dimensional embedded sub-...
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### Integral of a differential form on a torus

Can someone check if my work here is on the right track: Compute $$\int_{S^{1} \times S^{1}} f^{*} \mathbb{\omega}$$ where $f\colon S^{1} \times S^{1} \rightarrow \mathbb{R}^{4}$ is a smooth map ...
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### De Rham cohomology of $S^1$ with compact supports (Bott/Tu)

This is a question about Example 2.9, in Bott/Tu - Differential Forms in Algebraic Topology. Consider the decomposition of $S^1=U\cup V$ by two open sets, as in the figure above. Then both $U$ and $V$...
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### Showing that the differential is an immersion

If $f: X \rightarrow Y$ is an immersion of smooth manifolds, then show that $df: TX \rightarrow TY$ is also an immersion. The definition of immersion(when dim$X <$ dim$Y$) that I have is that for ...
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### Examples of no-zero-measure meagre set

I know cantor set and rational numbers in $\mathbb{R}$ are meagre. But they are all zero measure. So is there any meagre set that is non-zero measure?
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### On every metric space $(T, d)$ metric $d:T^2 \to \mathbb{R}$ is continuous function? [duplicate]

On every metric space $(T, d)$ metric $d:T^2 \to \mathbb{R}$ is continuous function? How to prove it?
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### The vertex of a tetrahedron is homeomorphic to a disk. [closed]

I want to prove that the tethahedron is a 2-manifold, but I have difficulties when it comes to deal with the vertices. It's natural that they are homeomorphic to a disk, but I find it hard to prove.
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### Understanding Hessian on Manifold (without Riemannian Geometry)

I've been going through notes on Morse theory and Handlebody theory and I've been having some trouble with the definition of the Hessian provided. The notes are on pages 3-4 here http://people.math....
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### Showing that all fixed points of a homotopy are isolated - possible using fixed point index?

I have a function $F(X,y)$, with $F: [0,1]^n \times [0,\infty)\rightarrow[0,1]^n$ i.e. essentially a self-map on a compact subset of $R^n$ with one parameter $y$. $F$ ist real-analytic. I know that ...
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