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Questions tagged [fibration]

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

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induced map in homology on a fiber bundle

Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map $f: E \rightarrow E$ that covers a map $g: B \...
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2 out of 3 property of fibrations?

Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a diagram in the category of topological spaces. If $g\circ f$ and one of $f,g$ are fibrations, can we conclude that so is the other? In this question it is ...
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Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
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Is $\mathbf{Cat}/\mathcal{C}\simeq\mathbf{Cat}^{\mathcal C^\mathrm{op}}$ when $\mathcal C$ is a $1$-category?

Given a small set $S$, we can define the overcategory $\mathbf{Set}/S$ to be the category whose objects are pairs $(A:\mathbf{Set},a:A\to S)$ and whose morphisms $(A,a)\to(B,b)$ are functions $f:A\to ...
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Integrable system vs lagrangian fibration

Every complete integrable system $I:M \rightarrow \mathbb{R}^n$ is a regular langrangian fibration on a dense subset of the symplectic manifold $M$. It is also known that locally every lagrangian ...
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1answer
48 views

Surjective homotopy equivalence which is not a fibration?

This is probably obvious to topologists so I'll just come right out with the question: What is an example of a surjective homotopy equivalence $E \to B$ of path-connected CW complexes which is not a ...
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About a Kan fibration (Postnikov towers for simplicial sets)

I am trying to understand a specific construction of Postnikov towers for simplicial sets, as explained for instance here (under "absolute Postnikov tower") So you start with a simplicial set $X$ (I ...
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Given a finite index normal subgroup of a fundamental group, how to construct a finite etale cover over a smooth projective curve?

Let $S,C$ be smooth projective surface,curve over the complex number field $\mathbb{C}$ ,respectively. Let $f:S\to C$ be a smooth fibration with general fiber $F$.Let $g=g(F)\geqslant 1$ and $n\...
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How to see link of a divide given in A'campo's paper is link

In the paper "Generic immersions of curves,knots,monodromy and gordian number" by A'Campo definition of a divide and divide of link given as below First of all as P is immersion it is not have to be ...
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Wang exact sequence with base space homology sphere

Let $F\rightarrow E\rightarrow S^n$, $n\geq 2$, be a fibration. Then we have the Wang exact sequence, $$ \cdots\rightarrow H_q(F)\rightarrow H_q(E)\rightarrow H_{q-n}(F)\rightarrow H_{q-1}(F)\...
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Nontriviality of the Hopf Fibration

A simple question how to understand why even though locally $S^3$ is homeomorphic to $S^2\times S^1$, how do you see that globally this is not true?
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Model category of all model categories

Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What ...
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Limits in a Grothendieck fibration

$\newcommand{\E}{\mathcal{E}} \newcommand{\B}{\mathcal{B}}$ I'm currently studying a paper that talks a lot about Grothendieck fibrations and so I'm trying to work with them a bit to get used to them. ...
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Relation between transport functor of a fibration and a Hurewicz connection on it

Let $A\overset{\alpha}{\rightarrow}B$ be a (Hurewicz) fibration. The homotopy lifting property w.r.t a fiber $\alpha ^{-1}(b)$ furnishes for each path $b\to b^\prime$ in the base a continuous map $\...
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degeneracy of the Serre spectral sequence

The following are well-known facts on the Serre spectral sequence For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field ...
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Replacing $H$-space multiplication with a fibration

Let $X$ be an $H$-space with multiplication $\mu: X \times X \to X$. Does there exist a space $\overset{\sim}{X}$ homotopy equivalent to $X$ such that the induced map $\overset{\sim}{\mu}: \overset{\...
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Doubts on obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
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1answer
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Fibration with contractible total space

Let $F \to E \to B$ be a fibration, and suppose $E$ is contractible. Then the long exact sequence of homotopy groups shows that $F$ has the same homotopy groups as the loop space $\Omega B$. Is it ...
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Fiber sequence, a group and an $n$-group

Given a short exact sequence $$ 1 \to B\mathbb{Z}_2 \to \mathbb{G} \to O(n) \to 1 $$ and the fiber sequence: $$ B^2\mathbb{Z}_2 \to B\mathbb{G} \to BO(n), $$ classified by $\beta \in H^3(BO(n), \...
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Cobordant of Dold manifold and Wu manifold via fibered classifying spaces

Background: I think, Dold manifold and Wu manifold are 5-dimensional manifolds which are cobordant to each other via 5-dimensional bordism group: $$ \Omega^{SO}_5. $$ Literally, cobordism theories ...
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Showing that this map descends to the quotient in an injective way

Let $f : \mathbb{S}^3 \to \mathbb{S}^2$ be the map $$ f(z_1,z_2) = (2z_1 \overline{z_2}, \vert z_1 \vert^2 - \vert z_2 \vert^2), $$ where we regard $\mathbb{S}^3 \subset \mathbb{C}^2$ and $\mathbb{...
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Surjectivity of Hurewicz fibrations

Proposition 4H.1 in Hatcher's Algebraic Topology states that any cofibration is injective. Since cofibrations and fibrations are dual concepts (in a way that I can't yet state formally, though), this ...
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Understanding the Hopf Link

I am trying to understand why the preimages of two points under the Hopf fibration are linked. I thought that two circles in $\mathbb{C}^n$ are linked iff one circle intersects the convex hull of the ...
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1answer
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Fundamental group about fibration

Let $E\to B$ be a fiber bundle (or more genereal, a fibration), and $B'$ be a section. Is it true that $\pi_1(B')$ is a subgroup of $\pi_1(E)$? Is it true that $\pi_1(B)$ is isomorphic to $\pi_1(B')$?...
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Mapping Tori and Monodromy

I have a question regarding the right setup for mapping tori. Let me give the definitions that I use first. Let $I$ denote the interval $[0,2\pi]$ and let $I^*$ denote the quotient of $I$ by the ...
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Open book on any closed oriented 3-manifold

I am reading "Lectures on Open Book Decompositions and Contact Structures" by Etnyre. In that paper Etnyre proves existence of open books on closed oriented 3-manifolds by using two lemma's of ...
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Topology and groupoids, proof 7.2.5 lemma.

Pg 269, 7.2.5 lemma Let $i:A \rightarrow X$ be a cofibration. Let $H:X \times \mathbb{I} \rightarrow Y$ be a homotopy $f \simeq g$, and let $\mathcal{G}$ be a homotopy rel end maps $G=H(i \times 1)$ ...
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When the primary obstruction is the only obstruction, is it still the only obstruction after a pull-back?

Consider the following situation: We have a Hurewicz fibration $p: E \rightarrow B$ with path-connected base $B$ and $(d-1)$-connected fiber $F$ for some $d \geq 1$. In case $d=1$ we require $\pi_1(...
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Collapse of Serre spectral sequence in the presence of a cross-section

I was under the impression that if a Serre fibration $f: E \rightarrow B$ has a right inverse $s: B \rightarrow E$, then the associated Serre spectral sequence would collapse on the second page. This ...
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When is the restriction of a fibration to a subspace a fibration?

Let $p:E\rightarrow B$ be a fibration and $E_0\subset E$ a subset. When is the restriction $p_0=p|:E_0\rightarrow B$ also a fibration? It seems like there should be some simple conditions which ...
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1answer
37 views

Integer Cech cohomology of $SU(3)$

I am interested in computing the Cech cohomology (with integer coefficients) of the group $SU(3)$. I particularly care about $H^k(SU(3),\mathbb{Z})$ with $k=7$, although ideally I would like to be ...
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Fibrations over contractible spaces

A fibration $\pi : E \to B$ over a contractible space is fiber-homotopy equivalent to the trivial fibration $B \times \pi^{-1}(b)$ for any point $b \in B$. A fiber bundle over a paracompact space is a ...
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How to define an action of $\pi_1(E)$ on $\pi_n(F)$ for a fibration $F\to E\to B$?

Given a fibration $F\to E\to B$, how to show that the action $\pi_1(F)$ on $\pi_n(F)$ can factor through $\pi_1(E)$: $\pi_1(F)\to\pi_1(E)\to\text{Aut}(\pi_n(F))$? This is exercise 4.3.10 from Hatcher'...
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The canonical null-homotopy of the fiber sequence $F(f)\to X\to Y$.

Let $F(f)$ be the homotopy fiber of the map $X\to Y$. Then there is a fiber sequence $F(f)\xrightarrow{p}X\xrightarrow{f}Y$, where $p:(x,\gamma)\mapsto x$. I wonder why the map $fp$ is (pointed) null ...
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1answer
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How to prove that $U(n)/O(n)\rightarrow S^1$ is a fibration?

I have a short question concerning the answer to this question. There it is used that the maps $$U(n)/O(n)\longrightarrow S^1,\quad A\longmapsto \det(A)^2$$ as well as, later, $$SU(n)\longrightarrow ...
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Example of a Serre fibration between manifolds which is not a fiber bundle?

I'm looking for an example of a map $f : X \to Y$, where $X$ and $Y$ are manifolds (without boundary), and $f$ is a Serre fibration, but $f$ is not a fiber bundle. I know that if $f$ is proper, and $...
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1answer
51 views

Characterisation Fibration

I have a question about a remark which provides an equivalent characterisation for (weak) Serre fibration: By "classical" definition a map $p:X \to Y$ is a (weak) Serre Fibration if it has following ...
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Low-degree integral cohomology of $K(\mathbb{Z}/n,2)$

Consider the Serre spectral sequence for the fibration $K(\mathbb{Z}/n,1)\rightarrow * \rightarrow K(\mathbb{Z}/n,2)$, $$ E^{pq}_{2}=H^{p}\bigl(K(\mathbb{Z}/n,2);H^{q}(K(\mathbb{Z}/n,1);\mathbb{Z})\...
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A disk bundle whose projection is not a homotopy equivalence

It's well known that with CW hypotheses, the projection of a disk bundle is a homotopy equivalence. But no one seems to make this claim in general, which leads me to believe it might, somehow, be ...
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Example of a fibration which does not arise from a fibre bundle

A fibration, $p\colon E\to B$ and a continouos map $\pi \colon F\to B$ , and a homotopy equivalence between $E$ and $F$ which respects the fibres gives $\pi \colon F\to B$ the structure of a fibration....
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On quotient space and fiber bundle

Let $G$ be a topological group acting on a space $X$, what conditions do I need to impose on $G$, $X$ and the action for the map $X\rightarrow X/G$ to be a (Hurewicz) fibration? Moreover, what ...
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Fibration with fibre $\mathbb{C}P^\infty$

I have a fibration $F\to E\to B$ where $B$ is a nice space (a compact manifold) and the fibre is $\mathbb{C}P^\infty$, i. e. an Eilenberg–MacLane space $K(\mathbb{Z},2)$. Are there good criteria to ...
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local system of coefficients on a fibration of classyfing spaces

It is well known that if $G$ is a lie group and $H$ is a closed subgroup of $G$, the inclusion $H \hookrightarrow G$ induces a fiber bundle on the classifying spaces $$ G/H \rightarrow BH \rightarrow ...
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Prove a function is a weak fibration.

From Rotman's Algebraic Topology: I'm having trouble proving that this is a weak fibration. I can see that diagonal map for the commutative diagram of a fibration must be of the form $\tilde G(\bar ...
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1answer
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group action on a fiber bundle

I am in the following situation, Let $F \rightarrow E \xrightarrow{p} B$ be a fiber bundle and suppose that we have a group $G$ acting freely on $E$ and $B$, satisfying $p(g\cdot x) = g \cdot p(x)$; ...
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1answer
155 views

interpreting a long exact sequence of homotopy groups

It is a well-known result in homotopy theory that a fibration $F \rightarrow E \rightarrow B$ induces a long exact sequence in the homotopy groups; namely, $$\pi_n(F) \rightarrow \pi_n(E) \rightarrow ...
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1answer
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Hodge numbers of base of fibration

Let $X$ be a smooth complex manifold which is a fibration over a complex base manifold $B$. Supposing we know the details of the fibration, is it possible from knowledge of the Hodge numbers of $X$ to ...
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1answer
157 views

What is the Eilenberg-MacLane space $K(\mathbb Z_2, 2)$?

Question 1: I know that $K(\mathbb Z, 2)$ is $\mathbb CP^\infty$ and that $K(\mathbb Z_2, 1)$ is $\mathbb RP^\infty$. But, how about $K(\mathbb Z_2, 2)$? Do we similarly have a good description of ...
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Cohomology of genus g surface fibration

Let $F\to E\to X$ be a Serre fibration with connected base space $X$ and let the fibre $F$ be an orientable surface of genus $g$. Is there anything we can say about how $H^n(E)$ depends on $H^n(X)$? ...
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Are topological fiber bundles on the same base with homeomorphic fibers isomorphic?

Let $\pi \colon E \to B$ and $\pi' \colon E' \to B$ two topological fiber bundles on the same base $B$. A $B$-morphism $f \colon E \to E'$ is an isomorphism of fiber bundles iff for all $b \in B$ the ...