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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 ...
marc's user avatar
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Computation of the Rings $H^*(K(\mathbb{Z}, n) ; \mathbb{Q})$

I'm studying Fomenko-Fuchs lecture 26.2 and I'm studying the following Theorem at page 371: Theorem. $$ \text{n odd} \implies H^*(K(\mathbb{Z}, n) ; \mathbb{Q})=\Lambda_{\mathbb{Q}}(x), \text{dim}x=n; ...
marc's user avatar
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1 vote
1 answer
60 views

Question about Whitehead Tower

I'm studying Miller lectures on Algebraic Topology, and I'm stuck in Theorem 68.9 ($\bmod \mathcal{C}$ Hurewicz Theorem ): Theorem 68.9 (Mod $\mathcal{C}$ Hurewicz theorem). Assume that $\mathcal{C}$ ...
marc's user avatar
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3 votes
1 answer
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$\bmod \mathcal{C}$ Vietoris-Begle Theorem

I'm studying Miller Lectures on Algebraic Topology, and I'm stuck in Proposition 68.7: Proposition 68.7 (Mod $\mathcal{C}$ Vietoris-Begle Theorem). Let $\pi: E \rightarrow B$ be a fibration such that ...
marc's user avatar
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0 answers
30 views

Geometric Morphisms and Fibrations in Higher Category Theory

I have been exposed to the idea of geometric morphisms in an ordinary categorical sense. There is a paper titled 'Fibred categories a la Jean Benabou' this somewhat links geometric morphisms and ...
Siyabonga Mthimkulu's user avatar
2 votes
1 answer
57 views

Spectral sequence of $\text { the fibration } E X \xrightarrow{\Omega X} X$

I'm studying Fomenko-Fuchs Lecture 22.3 and i'm stuck on the proof of the following theorem at page 333: Theorem: Let X be a topological space (with a base point), and let the space $X$ be $(n-1)$-...
marc's user avatar
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0 answers
39 views

If presheaves over C are generated by subterminal objects, is C is a posetal category?

I am trying to understand one of firsts statements in the proof of the lemma C5.2.4 of Johnstone's Sketches of an Elephant. It says that, for a small category $\mathcal{C}$, the existence of a ...
Dylan Facio's user avatar
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1 answer
46 views

How to understand May's proof that counit map is a weak equivalence?

A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
Christian's user avatar
2 votes
1 answer
86 views

Cohomology of $SO(3)$ bundle over $S^2$

To better understand spectral sequences, I tried to apply it for the following problem. Let's consider an $SO(3)$ bundle, $E$, over $S^2$. I know that there are two inequivalent $SO(3)$ bundles over $...
Tuhin Subhra Mukherjee's user avatar
2 votes
1 answer
64 views

Show that Mat$_{n\times k}^k\to G_k(\mathbb{R}^n)$ is a fibration

I am trying to show that Mat$_{n\times k}^k\to G_k(\mathbb{R}^n)$ is a fibration where Mat$_{n\times k}^k$ denotes the full-rank matrices of rank $k$. I know that $V_k(\mathbb{R}^n)\to G_k(\mathbb{R}^...
Michael Wang-Wakamatsu's user avatar
1 vote
1 answer
111 views

Error in Hatcher, Algebraic Topology?

In Hatcher's Algebraic Topology on page $409$ in the third paragraph it is written Given a fibration $p : E \to B$ with fiber $F = p^{−1} (b_0 )$, we know that the inclusion of $F$ into the homotopy ...
psl2Z's user avatar
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1 vote
2 answers
56 views

A sequence which is homtopy equivalent to a fibration is a fibration

Suppose we have a fibration $F\to E \to B$ and a sequence of maps $F'\to E' \to B'$ and suppose we have a map between these two sequences, which commutes up to homotopy and which is pointwise a ...
DevVorb's user avatar
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3 votes
0 answers
83 views

Gysin sequence from Serre spectral sequence vs Thom isomrophism

Let $n\geq 1$ and let $\pi:E\to B$ be a fiber bundle with fiber $S^n$ with $B$ simply connected. We will fix an orientation of $p$. An easy analysis of the Serre spectral sequence shows that there is ...
Ken's user avatar
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3 votes
0 answers
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Action of base space on homology of homotopy fiber, when the fiber is an Eilenberg Maclane space.

I am trying to understand a proof I found in a paper by Wagoner about delooping of algebraic K-theory (proposition 1.2 for those interested). For this I have a fibration $BE\to BG\to BG^{ab}$, with $E=...
DevVorb's user avatar
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2 votes
0 answers
57 views

Fibration coming from a group extension

I am trying to solve the following exercise about classifying spaces (5.1.28) in the book "Algebraic K-Theory and its applications" by Rosenberg: Let $$1 \longrightarrow N \longrightarrow G \...
The_Rookie's user avatar
3 votes
0 answers
49 views

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 ...
treutm14's user avatar
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0 answers
48 views

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 ...
Squirrel-Power's user avatar
2 votes
0 answers
25 views

Homotopy long exact sequence and image of connecting map in the center of some group [duplicate]

For context I am working on Weibel's K-book, chapter IV, my question comes from the proof of proposition 1.7. In this he claims that for the long exact sequence of a fibration with acyclic fiber $F\...
DevVorb's user avatar
  • 1,495
7 votes
2 answers
512 views

Question about isomorphism given by the Serre Spectral sequence

For context I am trying to reprove lemma 1.6 in Weibel's K-book chapter IV. The point about which I have question is the following. Suppose we have a fibration $F\to X\to Y$, with $F$ acyclic, then ...
DevVorb's user avatar
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2 votes
0 answers
65 views

Are principal fibrations the same as group bundles?

I am reading Hatcher's algebraic topology, where he defines a fibration $F\to E\to B$ to be principal if up to choices of homotopy equivalences it can be written as $\Omega B'\to F'\to E'\to B'$ (with ...
DevVorb's user avatar
  • 1,495
2 votes
1 answer
113 views

Every smooth fiber bundle admits an Ehresmann connection?

Is it true that every locally trivial fibration(fiber bundle) in smooth category admits an Ehresmann connection? The converse is obviously true, and it's one of the conclusion of the Ehresmann's ...
ChoMedit's user avatar
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1 vote
0 answers
16 views

A step in the Gromov's proof of contractability of $\omega$-tame complex structures on a finite dimensional vector space

Given a symplectic form $\omega$ on a symplectic vector space $V$, a complex structure $J$ on $V$ is said to be tamed by $\omega$ or $\omega$- tamed if $$\omega(v,Jv)>0$$ for all non zero $v\in V$. ...
Uncool's user avatar
  • 962
3 votes
1 answer
157 views

Homotopy fiber of the fold map

For context, my problem arises from "https://math.stackexchange.com/questions/4427532/homotopy-groups-of-wedge-of-spaces", where in one of the answers, the claim is that the homotopy fiber ...
DevVorb's user avatar
  • 1,495
2 votes
0 answers
80 views

Left divisor of a fibration by compact Lie group is a fibration.

Let $p:E \rightarrow B$ be a Hurewicz fibration where E and B are path-connected and compact CW complexes. Let $G$ be a compact Lie group (left) acting on $E$ and let $E’$ be the resulting orbit space ...
KFJ2611's user avatar
  • 153
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0 answers
66 views

finite covering of a torus bundle over the circle is a torus bundle over the circle

I am currently working through S.Wangs paper "On the existence of maps of nonzero degree between aspherical 3-manifolds" and got stuck at a part where he claims that for a closed orientable ...
Sanne J's user avatar
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Fourth Homotopy Group of $S^2$

I understand $\pi _{3}(S^2)$ by the Hopf Fibration which is the map $p(x_1,x_2,x_3,x_4)=(x_{1}^2+x_{2}^2-x_{3}^2-x_{4}^2, 2(x_1 x_4+x_2 x_3), 2(x_2 x_4-x_1 x_3))$. These are disjoint circle in $\...
Carlos Villeda's user avatar
1 vote
0 answers
36 views

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 ...
user38770's user avatar
  • 549
1 vote
2 answers
123 views

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 $C^{\rightarrow}$ ...
Mark's user avatar
  • 399
1 vote
0 answers
76 views

Steenrod squares in a Puppe sequence

Consider the following Puppe sequence. $$\cdots\to K(\mathbb{Z}/2\mathbb{Z},2)\to \Omega X\to K(\mathbb{Z}/2\mathbb{Z},1)\stackrel{a}{\to} K(\mathbb{Z}/2\mathbb{Z},3)\to X\to K(\mathbb{Z}/2\mathbb{Z},...
Leo's user avatar
  • 427
1 vote
1 answer
59 views

Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting

Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting? or even a UNIQUE homotopy lifting?cf parallel transport in diff geom
jim stasheff's user avatar
0 votes
2 answers
73 views

notation about $hfib(f,y_0)$ [closed]

Let $f:X\rightarrow Y$ be a continuous map. Then we can define $hfib(f,y_0)$. What does $hfib(f,y_0)$ mean?
Ziqiang Cui's user avatar
1 vote
0 answers
24 views

Nontrivial Monodromy of the Universal Stiefel Bundle (and $O(n)$-equivariant vector fields on spheres)

Note: I'm not allowed to embed images into my posts yet, so I've linked my diagrams instead. Throughout, we will make use of the following result. Fact. For $H$ a Lie subgroup of $G$, there is a ...
Baylee Schutte's user avatar
3 votes
2 answers
398 views

short exact sequence from fibration

Let $G$ be a finite group acting freely on a path connected topological space $X$. The covering map $X \to X/G$ induces a long exact sequence of homotopy groups. Since $\pi_1(G) = 1$ and $\pi_0(X) = 0$...
darko's user avatar
  • 1,265
0 votes
1 answer
86 views

Nullhomotopy implies existence of an arrow for homotopy fibration

Is it true that if $$F\to E\to B$$ is a homotopy fibration and $F\to X$ is nullhomotopic, then there exists a map $B\to X$ making the diagram commutative? Thank you.
Haldot's user avatar
  • 820
1 vote
0 answers
61 views

A couple of questions on homotopy fibrations. [closed]

I have several questions, but they all are mostly definition-centered and, I assume, are easy for a person with good understanding of fibrations (unfortunately, I am not one). Since there are several ...
Haldot's user avatar
  • 820
2 votes
1 answer
139 views

Homotopy equivalence of fibers and cofibers

Consider the following commutative diagram. $A$, $B$, $C$, $D$ are CW-complexes. $a$ and $b$ are homotopy equivalences. $\require{AMScd}$ \begin{CD} A @>f>> B\\ @VaVV @VVbV\\ C @>>g> ...
Leo's user avatar
  • 427
0 votes
0 answers
28 views

fibrations and hairbrush: an intuition

I would like to see why here it cannot be globally the case $E=B\times F$ and $\pi$ for the porjection in the hairbrush case ? Also what corresponds to $h,p,\tilde{h}_0$ in hairbrush.
user122424's user avatar
  • 3,978
0 votes
1 answer
50 views

the fibration in homotopy theory

I would like to see why here the name "fibration" has been chosen. What precisely are the fibres (threads) in this definition? The name is standard, but I have never seen its origin.
user122424's user avatar
  • 3,978
2 votes
0 answers
140 views

Action of $\pi_1(B)$ on Higher Homotopy groups $\pi_n(F)$ for a Fibration $F \to E \to B$

Let $F \to E \to B$ be a principal fibration, ie obtainable as a pullback from a path loop fibration $PX \to X$. Any path $\gamma: b_0 \to b_1$ in $B$ lifts to a homotopy equivalence $\overline{\...
user267839's user avatar
  • 7,439
1 vote
0 answers
77 views

When is the induced map in a pullback diagram a fibration?

Let $X$, $Y$ and $Z$ be topological spaces and let $f:X \to Z$ and $g:Y \to Z$ be continuous. Let $P$ be the pullback of $X\stackrel{f}{\to}Z\stackrel{g}{\leftarrow} Y$. Let $E$ be another space and ...
Stephan Mescher's user avatar
3 votes
0 answers
112 views

Homotopy Types of Fibrations in Postnikov Tower encoded by Classifing map

A Postnikov system of a path-connected space $X$ is an inverse system of spaces $$ \cdots \to X_{n}\xrightarrow {p_{n}} X_{n-1}\xrightarrow {p_{n-1}} \cdots \xrightarrow {p_{3}} X_{2}\xrightarrow {...
user267839's user avatar
  • 7,439
2 votes
0 answers
132 views

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 ...
Akerbeltz's user avatar
  • 2,733
1 vote
0 answers
100 views

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 $|q|:|X|\to|Y|$ ...
FShrike's user avatar
  • 42.5k
1 vote
0 answers
92 views

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 ...
Reznick's user avatar
  • 290
1 vote
2 answers
74 views

Difficulty with lemma $7.4$ of Goerss-Jardine: a simplex $\alpha$ is nullhomotopic to $v$ iff. there is a simplex with boundary $(v,\cdots,v,\alpha)$

We are given a nonempty fibrant simplicial set $X$ (a Kan complex) and $v\in X_0$ is any vertex. We are $\alpha\in X_n$ and the map $\alpha:\Delta^n\to X$ is homotopic to $v:\Delta^n\overset{!}{\...
FShrike's user avatar
  • 42.5k
1 vote
0 answers
45 views

How to find first nontrivial fibration in Whitehead tower of the $n$-sphere?

Consider the Whitehead tower of the $n$-sphere $X = S^n$: $$... \rightarrow X' \stackrel{p}{\rightarrow} X,$$ where $X'$ is $n$-connected. Explicitly, what is $X'$ and what is the fibration $p$? How ...
ccriscitiello's user avatar
5 votes
0 answers
242 views

Connecting map in the long exact sequence of homotopy groups

Let $p: E \to B$ be a Serre fibration, and given a basepoint $b_0 \in B$, let $e_0 \in p^{-1}(b_0)$ be a basepiont in the corresponding fiber. I'm having trouble proving that the connecting map $\...
Edmundo Martins's user avatar
1 vote
1 answer
57 views

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 ...
JLA's user avatar
  • 6,534
0 votes
1 answer
107 views

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 ...
galois1989's user avatar
2 votes
2 answers
142 views

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 ...
user1086467's user avatar

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