Questions tagged [fibration]

For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.

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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'...
Alvaro Sopeña's user avatar
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 $\mathbb{C}$ ...
Chicken feed's user avatar
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 ...
TopologyStudent's user avatar
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 \...
Yonatan Harpaz's user avatar
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 ...
Dan Ramras's user avatar
  • 8,498
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 \...
Jackson Walters's user avatar
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 $f$ ...
Andrea Marino's user avatar
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 $\Sigma_f X = X \times [0,1] / {\sim}$,...
Lauren's user avatar
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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 $M$ ...
return true's user avatar
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 $\operatorname{Map}(M,N)$ ...
Ken's user avatar
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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 ...
Mark Grant's user avatar
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 ...
JRoss's user avatar
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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 $A$, ...
Leo's user avatar
  • 541
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 ...
Lao-tzu's user avatar
  • 1,856
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 (...
Tanny Sieben's user avatar
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 $g:y\to x$ ...
Manuel Araújo's user avatar
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 ...
Leo's user avatar
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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 ...
Rylee Lyman's user avatar
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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 ...
Alec Rhea's user avatar
  • 8,947
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}^{...
Alec Rhea's user avatar
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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 ...
piper1967's user avatar
  • 1,059
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. ...
Alec Rhea's user avatar
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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 ...
user420620's user avatar
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-...
Chicken feed's user avatar
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, ...
Jeff Strom's user avatar
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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 ...
Mark Grant's user avatar
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 ...
user45397's user avatar
  • 2,195
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+...
Samuele's user avatar
  • 1,185
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 ...
IntegrableSystemsEnthusiast's user avatar
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 ...
user267839's user avatar
  • 5,938
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 ...
eta's user avatar
  • 53
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 ...
Alec Rhea's user avatar
  • 8,947
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 ...
user267839's user avatar
  • 5,938
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_{...
Ronald J. Zallman's user avatar
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 ...
Lukas Miaskiwskyi's user avatar
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 ...
qqqqqqw's user avatar
  • 915
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\...
E. KOW's user avatar
  • 732
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 ...
curious math guy's user avatar
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 ...
Math-Phys-Cat Group's user avatar
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 $1$-...
Aaron Maroja's user avatar
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/\...
Totoro's user avatar
  • 2,515
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 ...
Totoro's user avatar
  • 2,515
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 ...
Tina's user avatar
  • 373
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 ...
Freddie Manners's user avatar
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 ...
Qayum Khan's user avatar
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 ...
MKR's user avatar
  • 93
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{...
Hugo Chapdelaine's user avatar
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 ...
Ronald J. Zallman's user avatar
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 ...
Tyrone's user avatar
  • 4,981
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?...
Andrea Marino's user avatar