# Questions tagged [fibration]

A branch of topology that deals with the notion of a fiber bundle.

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### Question about the proof of Serre finiteness theorem

I'm studying the Serre finiteness theorem: Theorem The homotopy group $\pi_i\left(S^n\right)$ is finite for all $i$ except for $i=n$ and if $n$ is even for $i=2 n-1$, when it is finitely generated of ...
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### Nine lemma for fibration sequences

I am wondering if there is an analogue of the nine lemma for fibration sequences of spaces. The situation I am in is that I have the following diagram: All the columns and the first 2 rows are ...
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### Show that the submersions $V_k(\mathbb R^n) \to G_k (\mathbb R^n)$ and $O(n) \to G_k (\mathbb R^n)$ are fibrations

Show that the submersions $F: V_k(\mathbb R^n) \to G_k (\mathbb R^n)$ and $G: O(n) \to G_k (\mathbb R^n)$ are fibrations. Deduce that $\text{Mat}^k_{n \times k}(\mathbb R) \to G_k(\mathbb R^n)$ is a ...
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### Prove the category of all fibrations is a fibration.

Lemma 1.7.2 in Bart Jacobs' "Categorical Logic and type theory", asserts that the category of all fibrations $\mathbf{Fib}$ is a fibration over $\mathbf{Cat}$ via the codomain functor. A ...
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### The domain functor is a fibration?

I'm learning about fibrations and I read that the functor $dom: C^{\rightarrow} \rightarrow C$ is one for arbitrary category $C$. I can't see it. I need to show that any morphism in $C^{\rightarrow}$ ...
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### The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
• 2,733
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### Proving that the geometric realisation of a minimal fibration is a Serre fibration - have I got the details right?

$\newcommand{\O}{\mathcal{O}}$In the book: "Simplicial Homotopy Theory", by Goerss-Jardine, they 'prove' that every minimal fibration $q:X\to Y$ of simplicial sets $X,Y$ has $|q|:|X|\to|Y|$ ...
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1 vote
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### Fibres of a Hurewicz fibration are homotopic equivalent

Can someone provide me a proof or a reference of the fact that fibres of a Hurewicz fibration $E \to B$ are homotopic equivalent for a path-connected space $B$. This question has been asked before ...
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### Is this homotopy equivalence a deformation retract?

Suppose I have a Serre fibration of smooth manifolds $f:X\to Y$ (one may assume $Y$ is an open ball) with a section $s:Y\to X$ of $f\,,$ and furthermore assume the fibers of $f$ are contractible. This ...
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### Does this map satisfy the homotopy lifting property?

We have a $2-$dimensional disk with a segment attached in its center (like a plane umbrella). Let's call this space $T$. And consider the map $\pi: T\to\mathbb{D}^2$ which consists on projecting onto ...
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### Is the quotient map of an Abelian $G$-space a locally trivial fiber bundle without assuming compactness
Let $X$ and $G$ be locally compact, Hausdorff and second-countable spaces such that $G$ is an Abelian topological group with respect to that topology. Suppose that we have a continuous, free, and ...