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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2 answers
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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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$4$-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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Locally trivial fibration induced by a map

I am starting to study fibre bundles and I came across the following. If $(E,p,B,F)$ is a locally trivial fibration, and $g:B'\rightarrow B$ a map (continuous), then define $$g^\#E= \{(x,b') : p(x)=g(...
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4 votes
1 answer
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Is every fibration fiber homotopy equivalent to a fiber bundle?

A fibration $p : E \to B $ over a contractible base B is fiber homotopy equivalent to a product fibration $B \times F \to B$. (Corollary 4.63. Hatcher's Algebraic Topology) A locally trivial bundle(or ...
3 votes
1 answer
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Good reference for self study of Gauge theory [closed]

I am looking for the shortest way possible to study basic gauge theory. I am looking for some inspiring survey notes like this one: Christian Bär, Gauga Theory (rather than the great books by ...
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3 votes
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Are total spaces homotopy equivalent if base spaces are equal and fibers are homotopy equivalent in a fibration?

Suppose $B$ is path-connected, and admits the structure of a CW-complex. If there are two fibrations $F_1\rightarrow E_1\rightarrow B$ and $F_2\rightarrow E_2\rightarrow B$ such that $F_1$ and $F_2$ ...
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Calculating the intersection of $u^2 - v^3$ with a 3-sphere

For context, I'm coding a 3D visualisation of the Milnor fibration of a Trefoil knot. I've found some code https://www.unf.edu/~ddreibel/research/milnor/milnor-fibers.nb that calculates the ...
1 vote
1 answer
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Fibers of a locally trivial fibration are diffeomorphic

There is an "immediate" corollary in this paper that is not so immediate for me : https://people.math.osu.edu/george.924/Ehresmann%20Theorem This paper proves Ehresmann’s Theorem which ...
2 votes
1 answer
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Proving that all fibers of a morphism $\varphi:X\to\Bbb{P}^1$ are connected

Let $X$ be a rational elliptic surface over an algebraically closed field $k$ and $\pi:X\to\Bbb{P}^1$ its elliptic fibration, which I assume is relatively minimal. If $D$ is a nef divisor such that $D^...
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1 vote
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Serre fibrations and pullbacks

Let $h:Y \to B$ be a surjective Serre fibration and let the following be a pullback diagram. $$\require{AMScd} \begin{CD} X @>>> E @. \\ @VfVV @VgVV \\ Y @>>h> B @. \end{CD}$$ Then ...
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Taking $π:C^{n+1} \setminus \{ 0 \}→CP^n$, does the continuous application $q:CP^n→C^{n+1} \setminus \{ 0 \}$ exist such that $π∘q = Id$?

I'm trying to show that the continuous application $q:CP^n→C^{n+1} \setminus \{ 0 \}$ exists such that $π∘q = Id$ in $CP^n$. To show they are homotopy equivalent, I have been trying to use the lifting ...
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Equivalent Statements for the Fibre Bundle

Suppose $S$ is a Riemann Surface and $M$ is given as a $\mathbb{C} \mathbb{P} (1)-$bundle over $S$ where $ \pi:M \longrightarrow S$ is the corresponding projection. Is not the previous statement (...
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Examples of Seifert fibered homology spheres

The following is taken from chapter 1 of Saveliev's book Invariants for Homology 3-Spheres, there is the following definition: Let $a_1,\dots,a_n$ be positive integers, $n\geq 3$. Let $B=(b_{ij})$ be ...
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1 vote
1 answer
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Homotopy groups of infinite dimension lens space

Let $n > 1$. We define the infinite dimensional lens space $L$ as follows. Let $S^{\infty}$ be the unit sphere in the infinite dimensional complex vector space $C^{\infty}$, and let $\mathbb{Z}/n = ...
1 vote
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Definition of Principal Fibrations

$\require{AMScd}$ On page $412$, Hatcher defines a fibration $F\xrightarrow{}E\xrightarrow{}B$ is called principal if there is a commutative diagram \begin{CD} F@>{}>>E@>{}>>B\\ @VVV ...
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Showing evaluation map of rational function is a fibration

Let $Rat_d$ be the space of degree $d$ rational maps $f(z) = \frac{p(z)}{q(z)}$, where $p(z) = a_d z^d + \cdots + a_0, q(z) = b_d z^d + \cdots + b_0$ are complex polynomials with no common roots and ...
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1 vote
1 answer
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Using compactness in the proof of Fiber Bundles are Serre Fibrations

In the proof that fiber bundles have the homotopy lifting property for disks (i.e. it's a Serre fibration), Hatcher writes: I know how to use compactness to show we can subdivide $I^n\times I$ enough ...
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Fibration from homotopy orbits to classifying space

I was reading the answer to this question: https://mathoverflow.net/questions/836/do-homotopy-pullbacks-commute-with-homotopy-orbits-in-spaces It seems like if $X$ is a free $G$-space, then there is a ...
1 vote
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can $g(x)$ be associated to $\sum_n g(n^{-s})$?

While thinking about re-constituting real symplectic manifolds into complex ones via tori fibrations and mirror symmetry I thought about a possible association of objects: $g(x) \to \sum_n g(n^{-s})\...
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Based vs unbased fibration: Fiber sequence

My question is focused on basepoint issues in topology. The starting point is basically Problem 4.3.18 in Hatcher which asks us to show that a fibration sequence induces a long exact sequence after ...
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1 vote
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In May's book, the translation of fibers specifies a functor.

I am reading A Concise Course in Algebraic Topology written by J. P. May. On a page 53 of the book, he constructs the translation of fibers as follows: Let $p:E\rightarrow B$ be a fibration with ...
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Trivial fibration of semi-algebraic sets

Given a smooth bounded semi-algebraic set $M \subset \mathbb{R}^m$ and a projection $\pi:M \to \mathbb{R}^n$, let $K(\pi,M)$ denote the critical values of $\pi_{|M}$. We also require that the fibers ...
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Fibrations of manifolds with boundary - reference request

A smooth map $f\colon M\to B$ between compact manifolds is a fibration if it is surjective and submersive. I would like to use the same definition when $M$ and $B$ have boundary. Write $$ \partial M = ...
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How to prove that a map is a Hurewicz fibration?

A Hurewicz fibration is a continuous map $p:E\to B$ having the homotopy lifting property with respect to each space. Let X be a topological space. I want to prove that for all $n\geq 2$ the map $p_{n}:...
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Analog of a discrete fibration for sheafs?

Given a discrete fibration $ P : \text{Dis}(C^{\text{op}}) $ you can map to a copresheaf. $$ [[P]](x) = \Sigma s, C(P(s), x)$$ What is the equivalent if you want to map to a cosheaf on a site $(C, J)$?...
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Relationship between geodesics and dimension of spheres?

Let n be an odd positive integer and $U\subset S^{n}\times S^{n}$ be the open subset consisting of all couples (A,B) such that $A\neq -B$. Let $\pi:(S^{n})^{I} \to S^{n}\times S^{n}$ be the fibration ...
1 vote
1 answer
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Why is this a fibration (example 8, Spanier 2.4)?

This is example 8 from Spanier section 2.4. The example is as follows, we have $X$ to be the union of $$A_1 = \{(x, y)|x=0, -2\leq y\leq 1\}\\ A_2 = \{(x, y)|0\leq x\leq 1, y=-2\}\\ A_3 = \{(x, y)|x=1,...
1 vote
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construction of fibered $3$-manifold compatible with flat slices (fibered)

Consider a bounded (by a unit cube) real fibered $3$-manifold $M=(0,1)^3.$ Slicing $M$, with planes orthogonal to the faces of the cube, for example, $z=c$ for some positive constant $c\in(0,1)$ yield ...
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Zariski's connectedness theorem applied to $V\to B$ from a surface to a curve

Let all varieties over an algebraically closed field $k$ of arbitrary characteristic. Let $f:V\to B$ be a surjective morphism where $V$ is a smooth surface and $B$ is a smooth complete curve. I've ...
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Compute $\pi_{3+k}(S^3)$ with $k\le 3$ using Steenrod squares

As an application of the theorem (Serre) The cohomology ring $\mathcal{H}^*(K(\mathbb{Z}_2,n);\mathbb{Z}_2)$ is isomorphic to $\mathbb{Z}_2[Sq^I(\iota_n)]$ where $\iota_n$ is a fondamental class of $\...
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1 vote
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Regarding the Definition of Holomorphic Foliation on a Complex Manifold

Geometry, Dynamics And Topology Of Foliations: A First Course, Book by Bruno Scárdua and Carlos Arnoldo Morales Rojas, Chapter 1, Page 33. Definition 1.14. The definition of Holomorphic Foliation on a ...
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1 answer
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What does it mean that terminal object have morphism to other object?

Wiki defines terminal object as the object towards which there is a single morphism from every other object of the category https://en.wikipedia.org/wiki/Initial_and_terminal_objects This definition ...
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Inclusion of space loop is a Serre fibration

Let $i: X \longmapsto Y$ be an inclusion, and define $$\Omega_X Y:= \left\lbrace \gamma : I \longmapsto Y : \gamma(1) \in X \right\rbrace$$ Is it true that the map $$\bar{i} : \Omega_x Y \longmapsto Y$...
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$C^r$ Equivalent Fibre Bundles and Discrete Structure Group

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ and $(\mathbb{E}', \pi', \mathbb{B}, \mathbb{F})$ be two $C^r$ (where $r \ge 1$) equivalent Fibre Bundles, i.e., there is a $C^r$ diffeomorphism $H: \...
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2 votes
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Are loop space bundles fibrewise equivalent to principal bundles?

Suppose given a map $f:X\to Y$ with homotopy fibre a loop space $\Omega Z$. Then there is a homotopy equivalence $\Omega Z\simeq G$ for some topological group $G$. Does there exist a principal $G$-...
2 votes
1 answer
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Fibration with total space a symplectic manifold and base space connected such that fibres are symplectic submanifolds

In Dusa McDuff and Dietmar Salamon's book Introduction to Symplectic Topology, lemma 6.1.3(page 253) states that: Let $π\colon M\to B$ be a locally trivial fibration with connected base and $\omega \...
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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 ...
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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1 answer
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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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4 votes
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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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3 votes
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Fundamental group of the complement of some quadric cones

Problem Consider the domain $$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$ and the map $$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+z_3^2]\;.$$ Define $U=\...
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3 votes
1 answer
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Covering map associated to a fibration

I am reading about the generalization of the monodromy action to fibrations. In this settings, the lift of a path is not unique, but it is unique up to free homotopy. In particular there is a well ...
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4 votes
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Lefschetz Hyperplane Theorem's original proof

I'm trying to understand the main ideas used in the original proof by Lefschetz of his Hyperplane theorem. Here it is sketched shortly (source: Here) and I want to fill the gaps: Let $X$ be an $n$-...
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1 answer
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The end-point evaluation map in free path space is homotopy equivalence

Given $Y^I$ the free path space and a map $p:Y^I\to Y, \omega \mapsto \omega (1)$ it is to show that $p$ is a homotopy equivalence. I suppose this requires to show that the map $p$ is in fact a ...
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