Questions tagged [fibration]
For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
178
questions
38
votes
1
answer
5k
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Flatness in Algebraic Geometry vs. Fibration in Topology
I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic ...
38
votes
1
answer
2k
views
Is an affine fibration over an affine space necessarily trivial?
Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to $\...
30
votes
3
answers
2k
views
Is the counit of geometric realization a Serre fibration?
Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...
23
votes
2
answers
4k
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construct the elliptic fibration of elliptic k3 surface
Hi all,
As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?
...
20
votes
1
answer
804
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 ...
17
votes
6
answers
5k
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Cohomology of fibrations over the circle
Are there any general results on the (integral) cohomology of manifolds that are fibrations over the circle? Any literature references much appreciated.
17
votes
6
answers
2k
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A conceptual proof that local fibrations over paracompact spaces are global fibrations?
I'm teaching some homotopy theory at the moment, and I'm discovering a number of things I've never before learned properly. (I guess everybody who's ever taught knows that feeling.) One of those ...
15
votes
4
answers
4k
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What does it mean to speak of a homotopy fibration sequence?
I'm reading a paper in which the following is done. We have a certain particular map of spaces $f:X\to Y$ and then it is said something along the lines of "let $Z_f$ denote the space whose defining ...
15
votes
2
answers
1k
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Where does the primary obstruction of a fibration show up in its spectral sequence?
Let $f\colon\thinspace E\to B$ be a Serre fibration whose fibre $F$ is $k-1$-connected, $k\geq 1$. Assume $B$ is a connected CW complex. Then the primary obstruction to the existence of a cross ...
14
votes
3
answers
3k
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Multiplicativity of Euler characteristic for non-orientable fibrations
Let $E\to B$ be a fibration with fiber F, and assume for simplicity that B is connected. Suppose moreover that B and F have Euler characteristics (perhaps they are manifolds). Then often, one can ...
13
votes
2
answers
2k
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Request: A Serre fibration that is not a Dold fibration
A Serre fibration has the homotopy lifting property with respect to the maps $[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration $E \to B$ has the weak covering homotopy property: lifts with ...
13
votes
2
answers
858
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is there any fibration $\mathbb{R}^n\to \mathbb{S}^n$?
It is probably a trivial question. But I don't see the answer.
Is there any Hurewicz fibration $\mathbb{R}^n\to \mathbb{S}^n$ ?
Is there any fibration $X\to \mathbb{S}^n$, when $X\subset \mathbb{R}^...
13
votes
2
answers
2k
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Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\...
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 ...
12
votes
2
answers
799
views
global fibrations of simplicial sheaves
I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : ...
12
votes
1
answer
859
views
What is the difference between internal presheaves and presheaves on a total space?
Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$.
...
11
votes
3
answers
906
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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 ...
11
votes
1
answer
480
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 ...
11
votes
0
answers
263
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 $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
10
votes
2
answers
1k
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cup product and Steenrod operations in Serre spectral sequence
Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
10
votes
2
answers
822
views
Uniformization of Kodaira fibered surfaces
Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...
10
votes
2
answers
273
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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 $\operatorname{Map}(M,N)$ ...
10
votes
1
answer
498
views
Bundle-to-function correspondence
To a morphism of sets $f\colon E\to B$ with finite fibers, one may assign a function $$|f^{-1}|\colon B\to{\mathbb N}$$ sending an element $b\in B$ to the cardinality of the fiber $f^{-1}(b)$.
To a ...
9
votes
1
answer
2k
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How to Compute Transgressions in a Serre Spectral Sequence?
For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by ...
9
votes
4
answers
681
views
Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)
Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice $\...
9
votes
1
answer
989
views
Calabi-Yau fiber space without singular fibers implies finite quotient of product?
While reading this paper of Kollár, the following question came up. If $f:X\to Y$ is a fiber space (i.e. surjective holomorphic map with connected fibers) with $X,Y$ smooth projective manifolds,...
9
votes
1
answer
502
views
When is $X_{hG} \to X/G$ a weak equivalence for $X$ a free $G$-space, $G$ compact Lie?
Two questions (more details below):
Let $G$ be a compact Lie group and $X$ a $G$-space such that all stabilizer
subgroups are conjugate to a fixed $H \leq G$. Denote by $\pi: X
\to X/G$ the quotient ...
9
votes
1
answer
548
views
Homotopical interpretation of flatness?
I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:
A map of commutative algebras is flat if pushing it out ...
8
votes
2
answers
241
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 $\Sigma_f X = X \times [0,1] / {\sim}$,...
8
votes
1
answer
630
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}
&...
8
votes
2
answers
467
views
Constructively, are all fibrations cloven?
A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice".
Firstly, I'm a bit ...
8
votes
2
answers
862
views
Circle Action on Quaternionic Projective Space
Quoting from Wikipedia article on quaternionic projective space:
Therefore the quotient manifold
$$
\mathbb{HP}^{2}/\mathrm{U}(1)
$$
may be taken, writing $U(1)$ for the circle group. It has ...
8
votes
1
answer
558
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 ...
8
votes
1
answer
376
views
Elementary computation of direct image sheaves.
I am a physicist and would like to understand the section 1 of
this math paper, which explains how the SYZ conjecture implies topological mirror symmetry. I have some technical problem and would ...
8
votes
0
answers
148
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}, \...
8
votes
0
answers
593
views
Global geometry of discriminant locus
Let $X$ be a smooth projective threefold, and $\pi : X \to S$ an elliptic fibration over a surface (i.e. flat, with general fiber an elliptic curve). I'm interested in constructing such fibrations ...
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 ...
7
votes
1
answer
2k
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Quasi-unipotent monodromy for general families
This must be a naive question, but I'm wondering about the definition of quasi-unipotent monodromy for general families, not only 1-parameter families. The problem is that usually, in the books of ...
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.
...
7
votes
3
answers
884
views
A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
7
votes
1
answer
619
views
Where is simpleness used in the proof of existence of Postnikov towers of principal fibrations?
I've read one proof, rather long, in Allen Hatcher's book. There the key is Lemma 4.70, which uses the relative Hurewicz Theorem.
But there is another, shorter proof in J.P.May's book "A concise ...
7
votes
1
answer
491
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 ...
7
votes
1
answer
426
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-...
7
votes
1
answer
564
views
$\Pi$, $\Sigma$, and identity types without $\eta$ in comprehension categories
In comprehension categories, dependent sums are defined as a choice of left adjoints for all reindexing functors along display maps, satisfying a Beck-Chevalley condition. Dependent products are ...
7
votes
0
answers
639
views
Given a Serre fibration between manifolds, how ugly can it be?
A Serre fibration is clearly defined with motivation from homotopy theory, but we can consider smooth versions $f\colon M\to N$ in the category of (finite-dimensional, paracompact etc) smooth ...
6
votes
5
answers
2k
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Examples of Riemannian Submersions
Is there any example of a Riemannian submersion, which is no fibration?
As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. ...
6
votes
3
answers
3k
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The fiber of a Serre fibration
If $p:E\to B$ is a Serre fibration (assume it is surjective), then for each
$b\in B$ we get a comparison map $p^{-1}(b) \to F_b$, where $F_b$ is the homotopy
fiber of $p$ over $b$.
It is easy to ...
6
votes
2
answers
346
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 ...
6
votes
1
answer
823
views
Shafarevich conjecture for abelian varieties
In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties.
The statement is the following:
Let B be smooth projective a curve, S 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\...