Questions tagged [fibration]
For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
178
questions
0
votes
0
answers
39
views
How is the behaviour of a deformation retract under a fibration? [duplicate]
Let $p:E \rightarrow B$ a fibration and take $A\subset B$ a deformation retract of B. Is it true that $p^{-1}(A)$ is a deformation retract of E?
By deformation retract I mean the weaker definition.
I'...
6
votes
1
answer
280
views
Is this $\mathbb C$-fibration over compact Riemann surface trivial?
I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions:
$p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ ...
5
votes
0
answers
121
views
Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
4
votes
0
answers
91
views
Is the pushforward of an exponentiable fibration along an exponentiable fibration again exponentiable?
Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold:
The pullback functor $p^*\colon \...
20
votes
1
answer
803
views
Are all homotopy equivalences realized by fibrations over [0,1]?
Given two homotopy equivalent spaces $X$ and $Y$, does there always exist a Hurewicz fibration $p: E\rightarrow [0,1]$ with $p^{-1} (0) = X$ and $p^{-1} (1)=Y$?
This issue shows up in the accepted ...
1
vote
1
answer
617
views
Cohomology of the amplitude space of unlabeled quantum networks
I am investigating a particular map from a product of three-spheres to the moduli space of (non-negative, real edge weight) networks. The map in question is
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...
3
votes
1
answer
178
views
A fiber-like method to show equivalence of infinity categories
Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...
8
votes
2
answers
239
views
Is the fiberwise suspension of a Serre fibration a Serre fibration?
Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma_f\pi \colon \Sigma_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma_f X = X \times [0,1] / {\sim}$,...
2
votes
2
answers
184
views
Fibration of hyperbolic 3-manifold
A fibration of a manifold $\phi: M \to S^1$ gives rise to a short exact sequence
$$
1 \to \pi_1(N) \to \pi_1(M) = \mathbb{Z} \overset{f_\ast}{\to} 1
$$
where $N$ is the fiber.
I've heard that, if $M$ ...
10
votes
2
answers
273
views
Reference request - Fibrations between spaces of embeddings
This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
4
votes
0
answers
104
views
Monodromy action on homogeneous spaces
If $H$ is a Lie subgroup of $G$, then there is a fibration sequence
$$
G/H\to BH\to BG.
$$
By choosing a model for $EG$ we can promote this into a fibre bundle.
My question is about how to understand ...
3
votes
1
answer
160
views
Looking for examples of non-singular holomorphic foliations with compact leaves
I am looking for examples (or what is known about) of the following kind of object:
X compact Kähler manifold
F a non-singular holomorphic foliation on X (so given by a holomorphic subbundle of the ...
5
votes
2
answers
385
views
Associativity of consecutive fibrations
[ I asked the same question on stackexchange but attracted little attention. Besides, I made some progress after I posted it. So I decided to move it here. ]
Consider path-connected CW-complexes $A$, ...
6
votes
0
answers
186
views
(Co)cartesian fibrations and left Kan extensions
Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...
5
votes
1
answer
258
views
Computing the homotopy type of $B\operatorname{Aut}(K(G,1))$ using a fibration sequence: why is $G \to \text{Aut(G)}$ given by conjugation?
$\newcommand{\Aut}{\operatorname{Aut}}$Let $G$ be an abelian group.
It seems to be a well-known fact (for example here) that $B\Aut(K(G,1))$, the classifying space of the topological monoid of (...
4
votes
0
answers
150
views
Fibrations of $n$-groupoids in the folk model structure on $n$-categories
Define a strict $n$-groupoid to be a strict $n$-category all of whose morphisms are weakly invertible.
[For $1\leq k < n$ a $k$-morphism $f:x\to y$ is weakly invertible if there exists $g:y\to x$ ...
3
votes
0
answers
114
views
The $\pi_1(BM)$ action on $H^*(BH,R)$ in a Serre fibration $BH\to BG\to BM$
$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of ...
6
votes
1
answer
231
views
When is the Grothendieck / category of elements construction a fibration on geometric realizations?
Suppose we have a simplicial complex / poset / small category without loops $X$ equipped with a functor $F$ into the category of posets / small categories without loops. Suppose further that for each ...
3
votes
1
answer
344
views
Schemes as categories fibered in thin groupoids
Every time I start to read about schemes from a birds-eye view (like in the introduction to The Geometry of Schemes by Eisenbud and Harris) I get really excited; they sound like a categorical approach ...
2
votes
1
answer
135
views
Objects whose representable presheaf is a fibration
Is there any literature on representable presheaves which are fibrations, or categories such that all representable presheaves are fibrations?
A representable presheaf $$\mathcal{C}(-,X):\mathcal{C}^{...
3
votes
0
answers
71
views
Vietoris-Begle type result for differentiable fiber bundle
In Vietoris-Begle Theorem, we consider a closed and surjective map between two paracompact and Hausdorff spaces and we get some relation involving the homologies of the fiber, total space, and the ...
7
votes
1
answer
407
views
A fibration equivalent to having a terminal object
It is well known that the codomain functor $$cod:\mathcal{C}^\to\to\mathcal{C}$$ from the arrow category of a category $\mathcal{C}$ to itself is a fibration iff $\mathcal{C}$ has binary pullbacks.
...
3
votes
1
answer
204
views
Ehresmann's fibration theorem for CW or simplicial complexes
Is there an analogue of Ehresmann fibration theorem for (finite) CW complexes ?
Note is not true that an open surjective (necessary proper) cellular map of finite CW or simplicial complexes is ...
2
votes
1
answer
251
views
Complex fibration over complex torus
Let $M$ be a 3-dimensional complex manifold, and $\Lambda$ a discrete lattice in $\mathbb C^2$. Suppose there is a holomorphic submersion $f:M\to\mathbb{C}^2/\Lambda$ with fibers given by 1-...
1
vote
0
answers
135
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
4
votes
1
answer
349
views
fibre-preserving homotopy equivalence
Let $p:E\to B$ and $p':E'\to B$ be fibrations. It is well known that if $f:E\to E'$ a fibrewise map that is also a homotopy equivalence, then it is a fibrewise homotopy equivalence.
What about the ...
2
votes
0
answers
326
views
Trivialization of fibration by etale base change
Let $f:Y \to X$ be a smooth fibration over $\mathbb{C}$ in the sense that $X$ is a smooth, quasi-projective, connected variety and $f$ is a smooth, projective (surjective) morphism. Suppose that every ...
4
votes
0
answers
58
views
Fundamental group of the complement of some quadric cones
cross-posting from MathSE
Problem
Consider the domain
$$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$
and the map
$$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...
3
votes
0
answers
179
views
Integrable systems and Lagrangian fibrations
It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...
3
votes
0
answers
219
views
Historical proof of Leschetz Hyperplane Theorem
I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
...
5
votes
1
answer
254
views
Universal property of the codomain fibration
Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it ...
6
votes
1
answer
278
views
Can we show that a functor is a fibration without choosing a cleavage?
Is there a standard method for showing that a functor $F:\mathcal{C}\to\mathcal{D}$ is a fibration, aside from constructing a cleavage?
In the proof of the Grothendieck construction, the fibration we ...
3
votes
2
answers
2k
views
Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma
I have a general question about the motivation behind to definition the smooth morphisms
as we know it from algebraic geometry. The most common
definition of a smooth morphism $: X \to Y$ between two ...
3
votes
0
answers
193
views
Can we recover $\pi_2(S^2)$ from this simplicial set?
Let $S^3 \rightarrow S^2$ be the Hopf fibration. Can we recover $\pi_2(S^2)$ of $S^2$ from the simplicial set $X : \Delta^{op} \rightarrow \text{Set}$,
$$ X(n) = \pi_0 (S^3 \times_{S^2} \cdots \times_{...
3
votes
0
answers
64
views
Homotopy limits of section spaces
Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak ...
4
votes
0
answers
88
views
Free abelian group on a space and fibrations
Let $X$ be a topological space. Endow the free abelian group on $X$, $\mathbb Z[X]$, the quotient topology coming from the surjection $\bigsqcup_n X^n \times \mathbb Z^n \to \mathbb Z[X]$. For $Y$ a ...
6
votes
2
answers
1k
views
Action of fundamental group on homotopy fiber
For a Serre fibration of pointed topological spaces $f:X \to B$, there is an action of $\pi_1\left(B,b_0\right)$ on the fiber $F$. The construction of this action I'm familiar with uses a lift $F\...
4
votes
1
answer
800
views
Homotopy equivalent fibers and Fibrations
If a morphism of topological spaces $X\rightarrow Y$ is a fibration, and the target space is connected, then the fibers of the points $y\in Y$ are homotopy equivalent, i.e. for all $y_1,y_2\in Y$ we ...
6
votes
1
answer
445
views
CW structure on infinite-dimensional manifolds
It is well-known (due to this work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in this work) that they ...
1
vote
0
answers
162
views
Tischler's Theorem on nonvanishing $1$-forms on open manifolds
I have been trying to find a generalized version of the following theorem due to D. Tischler,
Theorem 1. Let $M^n$ be a closed $n$-dimensional manifold. SUppose $M^n$ admits a nonvanishing closed $1$-...
5
votes
0
answers
143
views
A fiber bundle of the Euclidean space over an orbifold
Consider a fiber bundle $p: F\hookrightarrow
E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...
7
votes
2
answers
328
views
Foliation of $\mathbb R^n$ by connected compact manifolds
Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not ...
12
votes
3
answers
1k
views
Classifying space for fibrations with Eilenberg-MacLane space fibers and nontrivial fundamental group actions
Let $A$ be an abelian group and let $n \geq 2$. For any connected CW complex $X$, it is standard that a fibration $f\colon E \rightarrow X$ whose fibers are homotopy equivalent to a $K(A,n)$ is ...
6
votes
0
answers
187
views
A notion of fibration on bisimplicial sets
[I am not trained in this stuff, but have an outside research interest, so sorry if this question is standard.]
I am interested in notions of fibrations, or fibrant objects, in bisimplicial sets. In ...
5
votes
1
answer
568
views
Delooping a fibration sequence with loopspace fiber and finite CW complexes
The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $K(G,n)$, and so I don't see the immediate ...
1
vote
0
answers
198
views
Zero Section on $\mathbb{P}^1$ Bundle
Suppose
\begin{eqnarray}
p: \mathbb{P}(V)\rightarrow S,
\end{eqnarray}
be a projective $\mathbb{CP}^1$ bundle. Is there any example, or is it possible at all, that the morphism $p$ doesn't have a ...
5
votes
1
answer
804
views
When is the cohomology of a fiber bundle a tensor product?
Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
1
vote
0
answers
116
views
Classifying Objects for Fibrations Defined by a Lifting Property
I have been studying weak factorization systems for their use in model categories. I am trying to use these to abstract away from a common phenomenon underlying fibrations. In brief, it seems as ...
7
votes
1
answer
490
views
Replacing the Fibre of a Fibration
This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature.
Let $p:E\rightarrow ...
0
votes
0
answers
130
views
"Smooth" Serre Fibrations (?)
Let $M,N$ be manifolds, $f:M \to N$ be a map.
In order to understand if $f$ is a serre fibration, it is enough to test it against differntiable maps $I^p \to M, I^{p+1} \to N$? What about smooth maps?...