# Questions tagged [fibration]

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

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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 ...
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### How to prove that a map is a Kan fibration

For simplicial sets $X$ and $Y$, let us denote by $\underline{\mathrm{Hom}}(X, Y)$ the simplicial set of morphisms $X \to Y$. If $p: X \to Z$ and $q: Y \to Z$ are morphisms of simplicial sets, let us ...
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### When is the quotient map of L(X) to its homotopy class of loops a fibration.

Let $L(X)$ be loop space of $X$ which is path connected surface (and its universal cover is contractible and $\pi_n=0$ for $n >1$ this details i'm giving as i'm working on such surface.), let $a$ ...
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### Can fibrations be described via universal morphisms?

Jacobs defines Cartesian morphisms and fibrations in Definition 1.1.3 of this document. In other places in the text he uses phrases like "the universal property of this lifting." This made ...
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1 vote
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### $4$-sphere as a $3$-sphere fibred over an interval

I have read that the following metric $$ds^2 = d\phi^2 + \sin^2(\phi)\,ds_{S^3}$$ (where $ds_{S^3}$ is the line element on $S^3$) “is the metric on $S^4$ written as a fibration of $S^3$ over the ...
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### Fibrations of $SU(4)$ and $SU(6)$

This MathOverflow answer gives a fibration of $SU(3)$ as a $S^3$ bundle over $S^5$. Are there similar fibrations for $SU(4)$ and $SU(6)$? And could you suggest books/resources where I can read up more ...
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### Betti number of a complex manifold which is a flat family

In practice, we wish to know if the topology of the base manifold and that of general fibres can somewhat "control" the topology of the total space. Precisely, let $\pi: X\to B$ be a flat ...
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1 vote
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### Fibration on arc-wise connected basis

Let be $(E,B;\pi)$ a fibration, with $B$ a topological spaces arc-wise connected. I know that $\pi$ is necessarily surjective and that for each path between two points in $B$ exists a path between two ...
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### What can be said about the homotopy groups of $(\widetilde{K}\times X)/G$

Let $G$ be a group and $X$ a simply connected $G$-space. For a $K(G,1)$ space $K$ with universal cover $\widetilde{K}\rightarrow K$ we have that $G$ acts on $\widetilde{K}$ via the unique homotopy ...
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### Proving that the Hopf Fibration is a fiber bundle

I'm having trouble understanding a proof that the Hopf Fibration is a fiber bundle. Here is the paper that I'm working through: hopf fibration Here is the paragraph from the paper that I am most ...
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### Serre spectral sequence and universal coefficient theorem

Let $F \rightarrow E \rightarrow B$ be a Serre fibration. Assume that the cohomological spectral sequence (with integer coefficients) $E_2^{p,q} = H^p(B; H^q(F)) \Rightarrow H^*(E)$ degenerates so ...
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