# 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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### When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
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### Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group, and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
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### How to Characterize Gradient Vector Fields?

Let $V$ be a vector field on a smooth manifold $M$. Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$? ...
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### Two Grassman manifold problems from Milnor and Stasheff's book

I'm stuck on the following two problems in Milnor and Stasheff's book Characteristic Classes. Really can't get my head around this material and I'm hoping that more worked examples would help. Even ...
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### The space of paths

Let $(M,g)$ be a compact Riemannian manifold, and let the path space $\Omega$ of $M$ be the set of equivalence class of smooth maps $\gamma : [0,1] \rightarrow M$ (equivalent under re-...
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### Which is harder to compute: $\pi_{n+k}$ or $\Omega^{fr}_n$?

Denote the $n+k$-th homotopy group of $S^n$ by $\pi_{n+k}(S^n)$ and the group of framed cobordism classes by $\Omega_n^{fr}(S^k)$. A central problem of algebraic topology is to compute $\pi_i(S^j)$ ...
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### Are closed orbits of Lie group action embedded?

Consider a smooth action $G\curvearrowright M$ of a Lie group on a smooth manifold. Suppose that an orbit $G\cdot p$ is closed. Is the orbit an embedded submanifold? In general we know that the ...
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### Extending metrics

Let $\pi:E\to M$ be a rank $k$ vector bundle over the compact manifold $M$ and let $i:M\hookrightarrow E$ denote the zero-section. Then we have a splitting of the restriction of $TE$ to the zero-...
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### Partial derivative of a composite function $\mathbb{R}^n \to \mathbb{R}^n$

I am trying to understand a proof but I am stuck on this technical bit: Apart from the small typo highlighted, I don't really see how to get the big formula for the partial derivative of $v_i$ ...
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### how to cover a manifold almost everywhere?

I pose the following question: Given a connected (maybe we need compactness) manifold or regular surface, can we find a single parametrization $\chi:B\to M$ from the open ball to the manifold, such ...
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### How to use the Prontrjagin-Thom construction to obtain the Gysin map?

I need help to understand the diagram in Miller's script Vector Fields on Spheres, etc. Chapter 23, p.82 on the bottom of the page. Before, Miller introduces the Prontrjagin-Thom construction: It is ...