Questions tagged [fibration]
A branch of topology that deals with the notion of a fiber bundle.
333
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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 $v\in V$. ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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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Steenrod squares in a Puppe sequence
Consider the following Puppe sequence.
$$\cdots\to K(\mathbb{Z}/2\mathbb{Z},2)\to \Omega X\to K(\mathbb{Z}/2\mathbb{Z},1)\stackrel{a}{\to} K(\mathbb{Z}/2\mathbb{Z},3)\to X\to K(\mathbb{Z}/2\mathbb{Z},...
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Showing $\text{fimm}(M,N)\to \text{Map}(M,N)$ is a fibration
I am trying to show $q: \text{fimm}(M,N)\to \text{Map}(M,N), (f,\delta f)\mapsto f$ is a Serre fibration.
Here $\text{fimm}(M,N)$ consists of pairs $(f,\delta f)$ such that $f: M\to N$ is a continous ...
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Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting
Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting? or even a UNIQUE homotopy lifting?cf parallel transport in diff geom
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notation about $hfib(f,y_0)$ [closed]
Let $f:X\rightarrow Y$ be a continuous map. Then we can define $hfib(f,y_0)$. What does $hfib(f,y_0)$ mean?
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Nontrivial Monodromy of the Universal Stiefel Bundle (and $O(n)$-equivariant vector fields on spheres)
Note: I'm not allowed to embed images into my posts yet, so I've linked my diagrams instead.
Throughout, we will make use of the following result.
Fact. For $H$ a Lie subgroup of $G$, there is a ...
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short exact sequence from fibration
Let $G$ be a finite group acting freely on a path connected topological space $X$. The covering map $X \to X/G$ induces a long exact sequence of homotopy groups. Since $\pi_1(G) = 1$ and $\pi_0(X) = 0$...
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Nullhomotopy implies existence of an arrow for homotopy fibration
Is it true that if
$$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.
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Definition of the map of homotopy fiber in a homotopy fibration
I remember the following definition of a homotopy fibration. If we have a continuous map $f:X\to Y$, then there exists a homotopy equivalence $h:X\to P_f$ and a (Hurewicz) fibration $p:P_f\to Y$ such ...
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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 ...
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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> ...
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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.
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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.
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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{\...
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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 ...
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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 {...
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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 ...
- 2,598
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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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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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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{!}{\...
- 35.3k
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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 ...
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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 $\...
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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 ...
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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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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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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 ...
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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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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 ...
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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 ...
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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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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 ...
user828175
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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 ...
user1028335
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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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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 = ...
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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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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 ...
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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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