The fibration tag has no usage guidance.

**7**

votes

**1**answer

234 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 ...

**5**

votes

**2**answers

328 views

### Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures?
Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...

**3**

votes

**3**answers

308 views

### Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?

This is a problem about the holomorphic fibration on a complex manifold.
Does there exist a holomorphic fibration of genus two over $\mathbb{CP}^{1}$ with 7 nodal singularities?
If you are aware of ...

**2**

votes

**0**answers

109 views

### Serre spectral sequence to compute cup product of bundles

Could we use Serre spectral sequence to compute cup product $\cup$ for the fibration
$$
F\to E\to S^m?
$$
My supervisor said this is true and wrote
$$
H^*(S^m)\to H^*(E)\to H^*(F),\\
\alpha\to ...

**4**

votes

**1**answer

174 views

### Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...

**5**

votes

**2**answers

246 views

### 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 ...

**4**

votes

**0**answers

137 views

### When is a circle fibration a circle bundle?

Let $\pi : E \to B$ be a Serre fibration over a CW complex, with circle fibers.
In the orientable case, it is easy to see that $\pi$ is fiber homotopy equivalent to a principal $SO(2)$--bundle.
...

**0**

votes

**0**answers

55 views

### Characterization of Singular locus

Let A be a complete regular local ring over a field k and B be a complete normal local ring over a field k. We assume that (Krull-dimension of A) > 1.
We consider the ring homomorphism f: A ---> B, ...

**2**

votes

**0**answers

49 views

### cohomology ring of cross-section space of one-point compactification of tangent bundle

Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. ...

**4**

votes

**2**answers

322 views

### Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...

**3**

votes

**1**answer

118 views

### group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...

**3**

votes

**1**answer

205 views

### Automorphisms of surfaces

Let $X$ be a projective surface with a morphism $f:X\rightarrow\mathbb{P}^1$. Assume that $f^{-1}(t)\cong\mathbb{P}^1$ for any $t\neq 0$ but $f^{-1}(0)$ is the union of two $\mathbb{P}^1$'s ...

**8**

votes

**2**answers

444 views

### 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: ...

**2**

votes

**0**answers

137 views

### A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
...

**9**

votes

**2**answers

284 views

### Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension).
Let $g:X\rightarrow ...

**3**

votes

**1**answer

251 views

### Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial

Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...

**2**

votes

**1**answer

142 views

### Fibration $p : \tilde Y \to Y$ with discrete fiber induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$

If $X$ is simply connected, locally path connected space and $p : \tilde Y \to Y$ is a covering map then it is easy to show that it induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$. Let's weak ...

**0**

votes

**0**answers

82 views

### How to understand/analyze vanishing cycles and fibers of 6 dimensional Lefschetz fibration?

Say you have a polynomial $f(x,y,z,m)=x^3+y^3+z^3+m^3$ where $x,y,z,m \in \mathbb{C}$. Consider the Lefschetz fibration from $\{f=\mu\} \cap \{ |m|\leq \delta \}$ to $m$ for suitably small $\mu$ and ...

**2**

votes

**0**answers

176 views

### Stable Lefschetz fibrations

Let $S$ be a non-singular complex projective surface, then construct the Lefschetz fibration $\pi:\widetilde S\longrightarrow\mathbb P^1$ associated to $S$ (we have a birational morphism ...

**1**

vote

**1**answer

175 views

### fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$.
...

**3**

votes

**1**answer

144 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

**1**answer

129 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

**2**answers

451 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 ...

**4**

votes

**1**answer

322 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

**2**answers

269 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 ...

**3**

votes

**0**answers

116 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

**1**answer

95 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

**0**answers

226 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 ...

**3**

votes

**1**answer

139 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

122 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

249 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 ...

**5**

votes

**4**answers

492 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

396 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

238 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 ...

**32**

votes

**1**answer

673 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 ...

**6**

votes

**2**answers

307 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

103 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

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

**23**

votes

**3**answers

808 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

157 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 ...

**1**

vote

**1**answer

221 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

390 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 ...

**3**

votes

**0**answers

265 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

311 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

293 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

313 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 ...

**3**

votes

**4**answers

372 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

649 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 ...

**4**

votes

**1**answer

705 views

### 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 ...

**3**

votes

**2**answers

294 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 ...