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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6
votes
2answers
354 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
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1answer
116 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 ...
5
votes
1answer
160 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
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0answers
59 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 $...
4
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0answers
80 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
2answers
250 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 ...
10
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3answers
680 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 ...
4
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0answers
115 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
1answer
296 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
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0answers
98 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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0answers
145 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
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1answer
240 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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0answers
107 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 ...
5
votes
1answer
315 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
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0answers
88 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?...
3
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1answer
197 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 ...
7
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1answer
235 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-...
1
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1answer
136 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 ...
8
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0answers
133 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}, \...
3
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1answer
199 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 ...
11
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0answers
214 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
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1answer
177 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 ...
4
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0answers
134 views

Torus bundle over spheres

I was wondering what is the classification of all torus bundles over spheres? That is, to classify the fibration $$ T^m \hookrightarrow M \to S^n. $$ It is well known that if $n=1$, all fibrations ...
2
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0answers
122 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 ...
11
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1answer
287 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 ...
4
votes
1answer
262 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 ...
2
votes
1answer
100 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 ...
5
votes
1answer
248 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 ...
6
votes
0answers
163 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
1answer
177 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 ...
4
votes
1answer
244 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 ...
5
votes
0answers
464 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 ...
2
votes
0answers
239 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
1answer
185 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}, \...
2
votes
0answers
61 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 ...
5
votes
0answers
89 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 \...
2
votes
0answers
82 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 ...
6
votes
1answer
452 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} &...
4
votes
1answer
152 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;% %...
2
votes
1answer
166 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 $F$...
8
votes
1answer
407 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 ...
1
vote
0answers
86 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 ...
4
votes
2answers
246 views

Space of sections of a fibration under weak homotopy equivalence

If I have two (Serre-)fibrations over the same base, and a weak equivalence of the total spaces that is also a map over the base, could I hope that the induced map on the spaces of sections would also ...
2
votes
1answer
145 views

Lefschetz Fibrations and disk bundles

When reading chaptes 7 of Akbulut's book about $4-$manifolds, he describes a handle decomposition for a manifold given a Lefschetz fibration over $S^2$. The idea is to extend the preimage of a disk ...
10
votes
3answers
672 views

Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?

((In conclusion) It was hard to choose which answer to accept. I decided for the one which addressed most of the various aspects of the question. ) (Later addon) I now decided to put a bounty on ...
5
votes
0answers
109 views

Cartesian liftings in double categories

The question: I wonder whether the following definition, or something similar, has appeared somewhere (see below for motivations). Any reference or pointer is welcome! (In what follows, I denote ...
6
votes
1answer
187 views

Simplicial approximation of a fibration

I am interested in the simplicial approximation of Serre or Hurewicz fibrations (or even fibre bundles). Let's assume $E$ and $B$ are finite simplicial complexes (or their associated geometric ...
4
votes
1answer
228 views

Lifting cellular structures to fibrations, fibre bundles or coverings

It is a well known result in Algebraic Topology that given a covering space $E\to B$ where the base has a CW-structure, then the total space can be given a CW-structure (see for example Theorem 8.10 ...
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vote
0answers
63 views

What is the other side of a category-indexed multicategory

By a category-indexed multicategory I mean a pseudofunctor from a category C to the 2-category of multicategories. For example, consider the pseudo functor which sends a topological space B to the ...
5
votes
0answers
150 views

Low dimensional homotopy fibration TOP(M) -> TOP(int(M))

In the thesis of Nancy Cardim she proves that for $M$ a topological manifold of dim $\geq 5$ with connected boundary, there exists a homotopy fiber sequence $C(\partial M)\rightarrow TOP(M) \...