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3
votes
1answer
101 views

What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...
2
votes
1answer
91 views

Open Books $( \Sigma, \Phi) $ living in Lefschetz Fibrations over the disk $D^2$

I have a question about open books and Lefschetz fibrations over the 2-disk $D^2$. Please let me set it up first, before going on. Setup: Say we have a Lefschetz fibration $f: W^4 \rightarrow D^2 $ ...
6
votes
2answers
338 views

Isotrivial fibrations over $\mathbb P^1$

First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard. $S$ is a smooth complex projective surface with a fibration $f$ over ...
3
votes
1answer
259 views

Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted: Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...
2
votes
2answers
208 views

A fibration of classifying spaces

Let $G$ be a Lie group, $N$ a closed connected normal subgroup. Let $BG$, $BN$, $B(G/N)$ be the classifying spaces of $G,N$ and $G/N$. Is there a fibration $BN\to BG\to B(G/N)$ ? It seems that such a ...
2
votes
0answers
97 views

Recognizing Simplicial (Quasi)Fibrations

Let's say we are given two finite simplicial complexes, which I will suggestively call $E$ and $B$. We'd like an algorithm for the following decision problem: Does there exist a simplicial map ...
2
votes
1answer
89 views

Going Back-and-Forth Between Different Expressions/“Representations” for Open Books.

I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
4
votes
0answers
130 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
1answer
116 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
1answer
106 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
1answer
229 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 ...
4
votes
4answers
345 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
2answers
331 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
1answer
224 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 ...
30
votes
1answer
513 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
2answers
244 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
0answers
96 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
1answer
343 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 $\mathbb{D}\to\mathbb{C}$. ...
22
votes
3answers
705 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
0answers
145 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
1answer
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
1answer
334 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
0answers
221 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
1answer
288 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
2answers
283 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
1answer
305 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 ...
2
votes
4answers
285 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
2answers
546 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
1answer
511 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
2answers
281 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
1answer
640 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
0answers
386 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
0answers
129 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
1answer
707 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
1answer
253 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
1answer
401 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
2answers
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
1answer
265 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
2answers
380 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
1answer
499 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
2answers
701 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
1answer
485 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
0answers
171 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
0answers
270 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
3answers
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 ...
11
votes
6answers
838 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
4answers
828 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
1answer
201 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 ...
3
votes
0answers
432 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
1answer
435 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 ...