# Questions tagged [fibration]

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

151
questions

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57 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 ...

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184 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
...

**4**

votes

**1**answer

113 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

189 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

491 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 ...

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166 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_{...

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45 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 ...

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75 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 ...

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13 views

### Partitioning $\mathbb{R}^3 into skew lines using hyperboloids, projectively transformed

There is a well-known fibration of projective space by the (nutually-skew) lines that rule a family of nested hyperboloids. E.g., the second image in https://en.wikipedia.org/wiki/Skew_lines. In ...

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votes

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618 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\...

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**1**answer

275 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 ...

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**1**answer

211 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 ...

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74 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 $1$-...

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93 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

280 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 ...

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votes

**3**answers

780 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 ...

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129 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 ...

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votes

**1**answer

349 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 ...

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vote

**0**answers

102 views

### Regarding the definition of $f$-morphisms/cartesian arrows in a fibred category $\mathcal{F} \rightarrow \mathcal{C}$

Let $p: \mathcal{F} \rightarrow \mathcal {C}$ be the data of a fibred category. Then, for arrows $f: U \rightarrow V$ in $\mathcal{C}$, a morphism $\phi: \xi \rightarrow \eta$ in $\mathcal{F}$ is said ...

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147 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 ...

**2**

votes

**1**answer

269 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{...

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108 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 ...

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**1**answer

342 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 ...

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103 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?...

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votes

**1**answer

235 views

### Serre and Hurewicz fibrations definition for pointed spaces?

I am a little bit confused by Serre and Hurewicz fibrations in the context of pointed spaces, i.e. in $Top_*$.
Serre and Hurewicz fibrations are defined in $Top$, i.e. for non-pointed spaces, as ...

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votes

**1**answer

278 views

### Examples of smooth Hurewicz fibrations which are not smooth fiber bundles

In the category of smooth manifolds (without corners), what are some examples of Hurewicz fibrations which are not fiber bundles?
The minimal topological example I know is to project the standard 2-...

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vote

**1**answer

190 views

### Relation between transport functor of a fibration and a Hurewicz connection on it

This is a crosspost of this MSE question.
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 ...

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135 views

### Mapping classes as Lefschetz fibrations over surfaces with positive genus

Let $\Sigma_{g,r}$ be the surface of genus $g$ and $r$ boundary components. It is known that, from a positive factorization of a mapping class $\phi$ in the mapping class group $MCG(\Sigma_{g,r}, \...

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**1**answer

219 views

### elliptic fibration over $\mathbb{P}^1$ with exactly two fibres with monodromy of unipotency rank 1

Despite the apparent simplicity of the following question I couldn't find the answer so far.
I am looking to construct an elliptic fibration $X \to \mathbb{P}^1$ with $X$ smooth, and exactly two ...

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225 views

### Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper

In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...

**2**

votes

**1**answer

200 views

### Leray-Serre spectral sequence for projective bundles

Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that ...

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143 views

### Restriction of the sheaf of relative differentials

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials.
For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...

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**1**answer

318 views

### Which maps of simplicial sets geometrically realize to fibrations?

If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...

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votes

**1**answer

274 views

### Does the Eilenberg Moore Construction Preserve fibrations?

Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$.
Then because the ...

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votes

**1**answer

105 views

### Naive compactification of $\mathbb{C}^*$-fibrations

Let $\pi:X \to Y$ be a $\mathbb{C}^*$-fibration between complex manifolds in the sense that there exists a fixed integer $a$ such that for every $y \in Y$, $\pi^{-1}(y)=(\mathbb{C}^*)^a$. Suppose ...

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**1**answer

263 views

### Half-dimensional torus fibration vs Lagrangian torus fibration

Assume we have a closed symplectic manifold $M$ which is the total space of a smooth fibration by half-dimensional tori. Can we infer that $M$ is the total space of a smooth fibration by Lagrangian ...

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172 views

### Degeneracy of the Serre Spectral Sequence

I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity.
In fact, for a Serre fibration $...

**6**

votes

**1**answer

190 views

### Symplectic Lefschetz fibrations in terms of factorization in symplectic mapping class group

There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma_g$ of genus ...

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votes

**1**answer

246 views

### Trivialization of Pontryagin square on oriented $4$-manifolds

I'm sorry for not clearly stating my question, thanks to Robert Bruner for answering my original question, let me restate it.
Let $\mathcal{P}:H^2(-,\mathbb{Z}/2)\to H^4(-,\mathbb{Z}/4)$ be the ...

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**0**answers

496 views

### Questions about 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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**0**answers

269 views

### Fundamental groupoid and fibration

In this post, it is said that a functor from the fundamental groupoid of a space $X$ (denoted by $\Pi(X)$) to the category $\mathrm{Vect}$ of vector spaces gives a flat vector bundle over $X$. But I ...

**5**

votes

**1**answer

216 views

### Transgression image and Serre spectral sequence for tori

Let $\mathbb{K} \subset \mathbb{T}$ be two tori acting on a topological space $X$ (with all the properties you want). We use the notations $$X_{\mathbb{T}} := (X \times E \mathbb{T}) / \mathbb{T}, \...

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65 views

### Condition for a map to carry over to Leray spectral sequences

I am trying to understand the conditions for two Leray spectral sequences to be related by a map.
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with ...

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95 views

### When is a bisimplicial set diagonal fibrant

Let $sSet^2$ be the category of bisimplicial sets.
In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \...

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94 views

### Extension of a given section and obstruction cocyles

Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a ...

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**1**answer

516 views

### Obstructions for the lifting problem after a pull-back

This is a cross-post from a MSE question which received no answers. Beware that the notation here is a little different.
Consider the following lifting problem(s):
$\require{AMScd}$
\begin{CD}
&...

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**1**answer

153 views

### The converse of Vietoris-Begle theorem

It is well known the following result:
Lemma: Let $F\rightarrow E\rightarrow B$ be a fibration with $B$ connected and
simply connected. Suppose that $F$ is $n$-acyclic, i.e. $H^{p}\left( F;%
%...

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votes

**1**answer

171 views

### Does a compact leaf of the smooth transversaly orientable foliation have trivial normal bundle?

In the book Geometric theory of foliations by Camacho and Neto, the following question is posed:
Let $G$ be a smooth transversaly orientable foliation. Let $F$ be a compact leaf of $G$. Prove that $F$...

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**1**answer

445 views

### About fibrations with fibre Eilenberg-MacLane spaces

Let $f: E\rightarrow B$ be a Kan fibration between pointed connected Kan complexes with fibre the Eilenberg-MacLane space $\mathrm{K}(M, n), n\geq 2, M$ an abelian group. Assume $f$ induces an ...

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87 views

### Transverse $S^1$ actions on mapping tori

Up until now I have thought that the existence of a transverse $\mathbb{S}^1$ action on a symplectic mapping torus implies that the mapping torus is trivial. Unfortunately I also came up with a ...