Questions tagged [fibration]
For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
64 questions with no upvoted or accepted answers
11
votes
0
answers
266
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 $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
- 1,329
9
votes
0
answers
120
views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
- 371
9
votes
0
answers
85
views
Reference for the tricategory of elements associated to a trifunctor
The theory of bicategorical fibrations has been relatively well studied, e.g. by Baković and by Buckley. In particular, given a trifunctor $F : \mathcal K \to \mathbf{Bicat}$ from a bicategory $\...
- 10.5k
8
votes
0
answers
153
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}, \...
- 1,409
8
votes
0
answers
608
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 ...
user47305
7
votes
0
answers
217
views
(Co)cartesian fibrations and left Kan extensions
Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...
- 1,906
7
votes
0
answers
172
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 ...
- 1,646
7
votes
0
answers
682
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 ...
- 35.4k
6
votes
0
answers
202
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 ...
- 915
6
votes
0
answers
300
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 $...
- 81
6
votes
0
answers
444
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.
...
- 599
5
votes
0
answers
133
views
Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
- 311
5
votes
0
answers
160
views
Fibrations of $n$-groupoids in the folk model structure on $n$-categories
Define a strict $n$-groupoid to be a strict $n$-category all of whose morphisms are weakly invertible.
[For $1\leq k < n$ a $k$-morphism $f:x\to y$ is weakly invertible if there exists $g:y\to x$ ...
- 291
5
votes
0
answers
156
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/\...
- 2,535
5
votes
0
answers
745
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 ...
- 521
5
votes
0
answers
124
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 \...
- 263
5
votes
0
answers
155
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) \...
- 93
4
votes
0
answers
81
views
Geometric morphisms for double categories
Is such there a notion 'Geometric morphism' for double categories? I have been reading Jean Benabou's lecture notes on fibred categories. He has slides on fibred geometric morphisms for arbitrary ...
- 615
4
votes
0
answers
191
views
Conduché fibrations
Given the following definition
Defn: Suppose $\mathcal{A}$ is a category with finite limits. An object $Y$ of $\mathcal{A}$ is called exponentiable if the functor $(- \times Y) \colon \mathcal{A} \to \...
- 615
4
votes
0
answers
117
views
Is the pushforward of an exponentiable fibration along an exponentiable fibration again exponentiable?
Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold:
The pullback functor $p^*\colon \...
- 9,481
4
votes
0
answers
111
views
Monodromy action on homogeneous spaces
If $H$ is a Lie subgroup of $G$, then there is a fibration sequence
$$
G/H\to BH\to BG.
$$
By choosing a model for $EG$ we can promote this into a fibre bundle.
My question is about how to understand ...
- 35.9k
4
votes
0
answers
63
views
Fundamental group of the complement of some quadric cones
cross-posting from MathSE
Problem
Consider the domain
$$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$
and the map
$$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...
- 1,205
4
votes
0
answers
203
views
Can we recover $\pi_2(S^2)$ from this simplicial set?
Let $S^3 \rightarrow S^2$ be the Hopf fibration. Can we recover $\pi_2(S^2)$ of $S^2$ from the simplicial set $X : \Delta^{op} \rightarrow \text{Set}$,
$$ X(n) = \pi_0 (S^3 \times_{S^2} \cdots \times_{...
user30211
4
votes
0
answers
92
views
Free abelian group on a space and fibrations
Let $X$ be a topological space. Endow the free abelian group on $X$, $\mathbb Z[X]$, the quotient topology coming from the surjection $\bigsqcup_n X^n \times \mathbb Z^n \to \mathbb Z[X]$. For $Y$ a ...
- 965
4
votes
0
answers
182
views
Naturality of primary obstruction under fiber-preserving maps
Let $B$ be a path-connected CW complex, and let $p:E\to B$ and $p':E'\to B$ be fibrations. Let $f:E'\to E$ be a fiber-preserving map, which therefore induces a map of fibers $\bar{f}:F'\to F$.
Let us ...
- 35.9k
4
votes
0
answers
295
views
Galois categories and the connected components functor
In stacks 0BMQ, a Galois category is defined to be a functor $F:\mathsf C\longrightarrow \mathsf{FinSet}$ such that $\mathsf C$ is finitely bicomplete, every object ...
- 10.5k
4
votes
0
answers
214
views
Some questions on Kontsevich's moduli space
Motivation: Work of Eisenbud, Harris, and Mumford shows that
$\mathcal M_g$ is of general type when $g≥24$. Moreover, by Logan's function $f(g)$ , $\overline {\mathcal M_{g,n}}$ is of general type for ...
user21574
3
votes
0
answers
118
views
The $\pi_1(BM)$ action on $H^*(BH,R)$ in a Serre fibration $BH\to BG\to BM$
$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of ...
- 663
3
votes
0
answers
78
views
Vietoris-Begle type result for differentiable fiber bundle
In Vietoris-Begle Theorem, we consider a closed and surjective map between two paracompact and Hausdorff spaces and we get some relation involving the homologies of the fiber, total space, and the ...
- 1,177
3
votes
0
answers
199
views
Integrable systems and Lagrangian fibrations
It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...
3
votes
0
answers
221
views
Historical proof of Leschetz Hyperplane Theorem
I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
...
- 5,998
3
votes
0
answers
66
views
Homotopy limits of section spaces
Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak ...
user126256
3
votes
0
answers
65
views
Name of the following "fundamental construction" on fibration
The following "fundamental construction" is mentioned in Bart Jacobs's thesis, from which we can derive a few useful fibrations
I have some blind spot ironing everything out, and wanted to look up ...
- 231
3
votes
0
answers
504
views
fibre sequence of classifying space
I read Steve Mitchell's Notes on principal bundles and classifying spaces (pdf).
There is a theorem: Let $G$ be any topological group, $H$ an admissible
normal subgroup. Then there is a homotopy-fibre ...
- 1,367
3
votes
0
answers
213
views
local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$
Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$.
To be more ...
- 543
3
votes
0
answers
176
views
Elliptic fibration arising from a higher genus linear system
Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$.
Let $L\subset H$ be a ...
- 502
3
votes
0
answers
177
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 $p:E ...
- 15.5k
3
votes
0
answers
367
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 ...
- 31
3
votes
0
answers
963
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 ...
- 14.4k
2
votes
0
answers
48
views
Double analogue of the domain functor
For a category $\mathcal{C}$, the domain functor $\mathbf{dom} \colon \mathcal{C}^2 \to \mathcal{C}$ is a fibration.
Denote $\mathbb{D}_1^2$ for the category of proarrows and cells between them and ...
- 615
2
votes
0
answers
106
views
Joyal's Cartesian squares
I am looking at Joyal's definition of cartesian squares Defn 1.1, Is this the same as saying that a commutative square is cartesian iff the induced arrows on the fibers are equivalences?
- 615
2
votes
0
answers
90
views
Equivalence of fibrations
I know that the for fibrations $P \colon E \to B$ and $Q \colon F \to B$ over $B$ there is an equivalence
$$\mathbf{Fib(B)}(R, P \to Q) \cong \mathbf{Fib(B)}(R \times_B P, Q) $$ natural in $R \in \...
- 615
2
votes
0
answers
69
views
Pullbacks in Cat in a 2-dimensional sense
$\newcommand\Fib{\mathrm{Fib}}\newcommand\Cat{\mathrm{Cat}}\newcommand\OpF{\mathrm{OpF}}\DeclareMathOperator\cod{cod}$In proving that a codomain functor from the 2-category $\Fib$ to $\Cat$ is a 2-...
- 615
2
votes
0
answers
379
views
Trivialization of fibration by etale base change
Let $f:Y \to X$ be a smooth fibration over $\mathbb{C}$ in the sense that $X$ is a smooth, quasi-projective, connected variety and $f$ is a smooth, projective (surjective) morphism. Suppose that every ...
- 2,323
2
votes
0
answers
327
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 ...
- 2,789
2
votes
0
answers
69
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 ...
- 1,227
2
votes
0
answers
158
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 ...
- 9,330
2
votes
0
answers
123
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 $M$. ...
- 2,223
2
votes
0
answers
314
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
$$
\...
- 14.1k
2
votes
0
answers
538
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 ...
- 21