The fibration tag has no wiki summary.

**4**

votes

**0**answers

97 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 ...

**2**

votes

**1**answer

109 views

### Equivalence of the total spaces of two Serre fibrations with equivalent fibers

Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such ...

**2**

votes

**1**answer

102 views

### modify a fibration with a fiber of higher multiple

Suppose we have an elliptic fibration $f:X\to \mathbb{P}^1$, with a singular fiber $F$, can we construct an elliptic fibration over $\mathbb{P}^1$ with fiber $nF$?

**6**

votes

**1**answer

217 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 ...

**3**

votes

**4**answers

325 views

### 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 ...

**8**

votes

**2**answers

316 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 ...

**5**

votes

**1**answer

217 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 ...

**29**

votes

**1**answer

441 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 ...

**5**

votes

**2**answers

238 views

### Why is the path fibration a strong Hurewicz fibration?

In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more ...

**0**

votes

**0**answers

94 views

### on the fibers over closed points

Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$
Let $f:X\rightarrow S$ a $k$-morphism of finite type.
We assume that for any closed point $s\in S(\bar{k})$, ...

**8**

votes

**1**answer

318 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 $\mathbb{D}\to\mathbb{C}$.
...

**19**

votes

**3**answers

611 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 ...

**0**

votes

**0**answers

143 views

### Change the fiber of a fibration

Let $F \rightarrow E \rightarrow B$ be a (Serre) fibration of topological spaces. Given a map $F' \rightarrow F$, is there a criterion for the existence or even an explicit construction of a fibration ...

**0**

votes

**1**answer

215 views

### special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which ...

**3**

votes

**1**answer

313 views

### Free Loops, Moore Paths and the Borel Construction

My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homotopy equivalent ...

**2**

votes

**0**answers

207 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 ...

**3**

votes

**1**answer

273 views

### Cohomology of a fiber bundle with fiber $H$ and base space $BG$

Are there any general results on the (integral) cohomology
of fiber bundle, where the fiber is a compact group $H$ (continuous or discrete)
and the base space is the classifying space $BG$ of another ...

**6**

votes

**2**answers

279 views

### Why is the base of SLAG fibration of CY3 expected to be $S^3$?

The SYZ conjecture roughly says that any Calabi-Yau threefold $X$ has a special Lagrnagian fibration $\pi:X\rightarrow B$ by 3-tous with section and one of its mirror partners $\check{X}$ is obtained ...

**8**

votes

**1**answer

302 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 ...

**1**

vote

**4**answers

265 views

### A homotopy equivalence between total spaces in a (Hurewicz) fibration which is not a fiber homotopy equivalence

In Hatcher's Algebraic Topology book it is noted after 4.61 that:
fiber preserving map + homotopy equivalence $\Rightarrow$ fiber homotopy equivalence.
Question:
Could there be two fibrations ...

**2**

votes

**2**answers

507 views

### Is every long exact sequence of homotopy groups induced by a fibration?

Is every long exact sequence
$$\cdots\to\pi_{d+1}(B)\to\pi_d(F)\to\pi_d(E)\to\pi_d(B)\to\pi_{d-1}(F)\to\cdots$$
with topological spaces $F,E$ and $B$, where $F$ is a subspace of $E$ with inclusion map ...

**3**

votes

**1**answer

444 views

### Quasi-unipotent monodromy for general families

This must be a naive question, but I'm wondering about the definition of the quasi-unipotent monodromy for general (not only 1-parameter families). The problem is that usually in the books of ...

**3**

votes

**2**answers

279 views

### Analogues of fibrations

Recall the following analogy
Serre fibrations : Kan fibrations
in spaces and simplicial sets respectively, related by the singular simplices functor and geometric realisation. There are other ...

**8**

votes

**1**answer

623 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 ...

**2**

votes

**0**answers

374 views

### Are all the smooth fibers in a fibration always homeomorphic?

Let $f:X \rightarrow Y$ be a fibration from a complex manifold $X$ to another connected complex manifold $Y$ such that all the fibers are compact, reduced, connected and smooth. Is it possible that ...

**1**

vote

**0**answers

123 views

### Classification of fibres in pencils of curves of genus two

For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.

**6**

votes

**1**answer

681 views

### 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 ...

**3**

votes

**1**answer

247 views

### How can I see that the slice of a presheaf category is equivalent to the presheaf category of the category of elements?

Let $\mathcal{C}$ be a small category and $P$ a presheaf on $\mathcal{C}$. Then there is an equivalence $$\widehat{\mathcal{C}} / P \cong \widehat{\int_{\mathcal{C}}} P$$ Now there are (at least) two ...

**1**

vote

**1**answer

389 views

### Computing the homology groups of spaces in a fibration

Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it ...

**5**

votes

**2**answers

1k views

### 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?
...

**3**

votes

**1**answer

255 views

### Terminology for fiberwise maps

I would like to know the standard terminology for the following two notions.
Notion 1: $E_1\to B$ and $E_2\to B$ are fibrations over the same base space, and $f\colon E_1\to E_2$ is a map making the ...

**1**

vote

**2**answers

378 views

### Codomain fibration.

It is known that the codomain fibration is given by a functor in the form $\mathcal{C}^{\rightarrow}\longrightarrow\mathcal{C}$ where $\mathcal{C}$ is a category having pullbacks and ...

**7**

votes

**1**answer

452 views

### 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 ...

**10**

votes

**2**answers

697 views

### 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 ...

**8**

votes

**1**answer

475 views

### 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 ...

**0**

votes

**0**answers

166 views

### Does there exist a fibration of genus two over P^1 with only 3 singular fibres but two are semi-stable fibers for algebraic surfaces?

We will work over the complex numbers C.
there exist a fibration of genus two over P^1 with only 3 singular fibres but one is semi-stable fiber.
there not exist a fibration of genus two over P^1 ...

**3**

votes

**0**answers

251 views

### Does there exist a non-isotrivial fibration of genus two over P^1 with only 3 singular fibres of general type surfaces?

We will work over the complex numbers C.
This question is based on Beauville's article :
there exist a non-isotrivial fibration of genus 2 over P^1 with only 3 singular fibres.
but not know for ...

**3**

votes

**3**answers

1k views

### 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 ...

**10**

votes

**6**answers

793 views

### 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 ...

**2**

votes

**4**answers

777 views

### 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. ...

**1**

vote

**1**answer

194 views

### Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces

Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.
Question: How do you prove that the following diagram of homotopy groups commutes?:
$\pi_n(Y) \to ...

**2**

votes

**0**answers

417 views

### How to prove that a map is a Serre fibration?

I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that ...

**9**

votes

**1**answer

432 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 ...

**2**

votes

**1**answer

771 views

### Question related to the moduli space of Riemann surfaces and a fibration

If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:
$M^1_{g} ...

**1**

vote

**0**answers

357 views

### Casson Gordon paper - Cobordism of classical knots

It is given in Progress in mathematics 62, Guillou and Marin book. In the proof of Lemma 4, They choose $\alpha$ and $r\in \mathbb{N}$ such that $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfies ...

**4**

votes

**1**answer

483 views

### Fibrations of Simplicial sets

Hello,
Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations ...

**11**

votes

**2**answers

952 views

### 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 ...

**26**

votes

**1**answer

2k views

### 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 ...

**7**

votes

**2**answers

423 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 : ...

**9**

votes

**6**answers

2k views

### 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.