Questions tagged [fibration]

A branch of topology that deals with the notion of a fiber bundle.

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3
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1answer
113 views

What does it mean that terminal object have morphism to other object?

Wiki defines terminal object as the object towards which there is a single morphism from every other object of the category https://en.wikipedia.org/wiki/Initial_and_terminal_objects This definition ...
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1answer
62 views

Inclusion of space loop is a Serre fibration

Let $i: X \longmapsto Y$ be an inclusion, and define $$\Omega_X Y:= \left\lbrace \gamma : I \longmapsto Y : \gamma(1) \in X \right\rbrace$$ Is it true that the map $$\bar{i} : \Omega_x Y \longmapsto Y$...
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$C^r$ Equivalent Fibre Bundles and Discrete Structure Group

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ and $(\mathbb{E}', \pi', \mathbb{B}, \mathbb{F})$ be two $C^r$ (where $r \ge 1$) equivalent Fibre Bundles, i.e., there is a $C^r$ diffeomorphism $H: \...
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Are loop space bundles fibrewise equivalent to principal bundles?

Suppose given a map $f:X\to Y$ with homotopy fibre a loop space $\Omega Z$. Then there is a homotopy equivalence $\Omega Z\simeq G$ for some topological group $G$. Does there exist a principal $G$-...
2
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1answer
32 views

Fibration with total space a symplectic manifold and base space connected such that fibres are symplectic submanifolds

In Dusa McDuff and Dietmar Salamon's book Introduction to Symplectic Topology, lemma 6.1.3(page 253) states that: Let $π\colon M\to B$ be a locally trivial fibration with connected base and $\omega \...
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30 views

No rational section to universal conic- topological obstruction analogue? Homology in algebraic geometry?

In Vakil, he has a very nice exercise 18.4.5 where we show that there are no 3 rational function in the coefficients of a conic (on $\mathbb{P}^2$) $a_{00},a_{01},a_{02},a_{11},a_{12},a_{22}$ called $...
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28 views

Fibrations of $SU(4)$ and $SU(6)$

This MathOverflow answer gives a fibration of $SU(3)$ as a $S^3$ bundle over $S^5$. Are there similar fibrations for $SU(4)$ and $SU(6)$? And could you suggest books/resources where I can read up more ...
3
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51 views

Betti number of a complex manifold which is a flat family

In practice, we wish to know if the topology of the base manifold and that of general fibres can somewhat "control" the topology of the total space. Precisely, let $\pi: X\to B$ be a flat ...
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54 views

Definitions of Isotrivial family and Isom-scheme

Let $f:S \to C$ be a fibered surface over smooth curve $C$ (defined over an algebraically closed field $k$.) In this discussion (Isotrivial family: different definitions) is asked if and when ...
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Fibration on arc-wise connected basis

Let be $(E,B;\pi)$ a fibration, with $B$ a topological spaces arc-wise connected. I know that $\pi$ is necessarily surjective and that for each path between two points in $B$ exists a path between two ...
2
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1answer
61 views

What can be said about the homotopy groups of $(\widetilde{K}\times X)/G$

Let $G$ be a group and $X$ a simply connected $G$-space. For a $K(G,1)$ space $K$ with universal cover $\widetilde{K}\rightarrow K$ we have that $G$ acts on $\widetilde{K}$ via the unique homotopy ...
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2answers
75 views

Proving that the Hopf Fibration is a fiber bundle

I'm having trouble understanding a proof that the Hopf Fibration is a fiber bundle. Here is the paper that I'm working through: hopf fibration Here is the paragraph from the paper that I am most ...
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49 views

Serre spectral sequence and universal coefficient theorem

Let $F \rightarrow E \rightarrow B$ be a Serre fibration. Assume that the cohomological spectral sequence (with integer coefficients) $E_2^{p,q} = H^p(B; H^q(F)) \Rightarrow H^*(E)$ degenerates so ...
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Fundamental group of the complement of some quadric cones

Problem Consider the domain $$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$ and the map $$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+z_3^2]\;.$$ Define $U=\...
3
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1answer
45 views

Covering map associated to a fibration

I am reading about the generalization of the monodromy action to fibrations. In this settings, the lift of a path is not unique, but it is unique up to free homotopy. In particular there is a well ...
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Lefschetz Hyperplane Theorem's original proof

I'm trying to understand the main ideas used in the original proof by Lefschetz of his Hyperplane theorem. Here it is sketched shortly (source: Here) and I want to fill the gaps: Let $X$ be an $n$-...
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1answer
29 views

The end-point evaluation map in free path space is homotopy equivalence

Given $Y^I$ the free path space and a map $p:Y^I\to Y, \omega \mapsto \omega (1)$ it is to show that $p$ is a homotopy equivalence. I suppose this requires to show that the map $p$ is in fact a ...
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102 views

Universal property of the codomain fibration

Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it ...
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1answer
82 views

Fibrations that are not fiber bundles

I noticed that this question has been asked before, but I haven't been able to find a good answer for it, so here goes. For a course project I am learning about fibre bundles and fibrations, and now I ...
4
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1answer
99 views

Is there a smooth homotopy lifting theorem?

Given a smooth map $p\colon E \to B$ of smooth manifolds with the continuous homotopy lifting property, does $p$ satisfy the smooth homotopy lifting property?
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can Chen's iterated integrals be used in Ehresmann's fibration theorem?

A proof in Dundas, Bjørn Ian, A short course in differential topology, Cambridge Mathematical Textbooks. Cambridge: Cambridge University Press (ISBN 978-1-108-42579-7/hbk; 978-1-108-34913-0/ebook). ...
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61 views

Action of fundamental group of the base space on the homology of the fiber

The following was extracted from Hatcher’s book of algebraic topology: Could you explain me with more detail what is this action of the fundamental group of the base space of a fibration $\pi: E \to ...
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1answer
54 views

A projection is a fibration

I recently learned about the concept of a Hurewicz fibration. I tried to prove that for a product of topological spaces $X \times Y$, the projection $p: X \times Y \rightarrow X$ is a Hurewicz ...
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27 views

For a fibration $p:E\rightarrow B$, if $b,b'\in B$ are in the same path component, then $F_b\simeq F_{b'}$

If $p:E\rightarrow B$ is a fibration (Hurewicz), and we take $b,b'\in B$ in the same path component, I want to prove that the fibers are of the same homotopy type $F_b\simeq F_{b'}$. I know I should ...
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0answers
18 views

Reindexing functors without a cleaving

Does it make sense to speak of reindexing functors for fibrations without a cleaving? A cleaving gives us concrete reindexing functors, but even without a specific cleaving we know that Cartesian ...
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1answer
61 views

Why is the Mapping Cylinder a fibering?

If $p:E \rightarrow B$ is a fibration with fiber $F$, in other words we have the fiber space $(E,B,F,p)$. Let $M_p$ be the mapping cylinder of $p$, $M_p = ((E\times[0,1]) \sqcup B)/\sim $, in which $(...
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0answers
37 views

For fibrations, is two out of three good enough?

Let $H_{X}\subset H_{Y} \subset H_{K}$ be closed subgroups of a topological group $G$. Let $X = G/H_{X}$ $Y = G/H_{Y}$, $K = G/H_{K}$. Now, suppose we have the following diagram: $$\require{AMScd}\...
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1answer
50 views

Geometrically seeing the linking of the fibres in the Hopf fibration?

I'm working through the following document: https://arxiv.org/pdf/0908.1205.pdf I can understand the construction of the Hopf map using the Riemann sphere but I struggle to understand the geometric ...
3
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2answers
128 views

Lifting a map to the total space of a circle bundle

Let $\pi:P \to M $ be a smooth circle bundle, so $S^1$ is the fibre, $f:N \to M$ a smooth map. I would like to know what are the necessary and sufficient conditions for $f$ to lift to a map $g:N \to P$...
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32 views

Seeing that the subobject fibration is a fibration

Let $\mathcal{B}$ be a category with pullbacks. We have that $cod:\mathcal{B}^\rightarrow\to\mathcal{B}$ is a fibration, with the Cartesian arrow over $k:X\to cod(g:Y\to Z)$ given by the pullback ...
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1answer
83 views

Is the map $z\mapsto z^2$ a fibration?

Let $f:\mathbb C \to \mathbb C$ be defined as $f(z) = z^2$. Is this a fibration? Or at least a Serre fibration? I am not sure how to approach this.
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1answer
80 views

generalization of fibration

A Hurewicz fibration $E\rightarrow B$ is a map so that for all maps $X\rightarrow E$ and $X\times I\rightarrow B$ making the obvious square commute, there is a lifting $X\times I\rightarrow E$ so that ...
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76 views

What is the essential difference between a sheaf and a fibration with lifting property?

Are there cases where the two notions coincide? By sheaf I mean a pre-sheaf ($\mathcal{C}^{op} \rightarrow \mathcal{Set}$) satisfying the sheaf condition. The sheaf condition says that, for every ...
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1answer
58 views

Equivalence of contractible and trivial fibrations

Let $B$ be a connected space. I am wondering, which of the following are equivalent, and under which conditions? $B$ is weakly contractible. $B$ is contractible. All fibrations $E \rightarrow B$ are ...
3
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1answer
101 views

Exact sequence of homotopy groups from a short exact sequence of Lie groups

Let $$ 1\to G'\to G\to G''\to 1 $$ be a short exact sequence of real Lie groups. I assume that $G',\,G$, and $G''$ have finitely many connected components, but are not necessarily connected. I am ...
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53 views

Up to homotopy principal bundle

A principal bundle (say, in the category of topological spaces and topological groups) induces homeomorphisms on the fibers. This works well for groups $G$. I am wondering if there is an analogue for &...
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0answers
55 views

What is a (co)fibration in category theory

Coming from a computer science background, I have some working knowledge on category theory but none in algebraic topology. I see many definition involves some homotopy theory which I completely have ...
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1answer
45 views

Model categories: $\text{Ho}$ and $\cal C_{cf}/\sim$

I have asked this question about model categories: Why $\text{Ho} \ \cal C$ is $\cal C_{cf}/\sim$ and not $\cal C/\sim$ and I got this answer: take for cofibratiobns Iso, weak equivalences all arrows. ...
4
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1answer
81 views

$\pi_2(T \vee \mathbb{C}P^2)$ and action of $\pi_1$ on $\pi_2$

Let $X= T \vee \mathbb{C}P^2,$ where $ T$ denotes the 2-dimensional torus. The task is to compute $\pi_2(X)$ and describe the action of $\pi_1(X)$ on $\pi_2(X)$. As for the first part, is there any ...
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1answer
53 views

natural isomorphisms

In the snippet below (taken from Hovey's MC book), I would like to understand in some detail the lemma 6.1.2 (what it says) and the proof of it. Its too sketchy for me. I do not even know what is Map$...
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2answers
49 views

Comparing the definition of a cone from Wikipedia and from Marty Arkowitz.

Here is the definition of a cone (on pg.76) from "Introduction to Homotopy Theory" by Martin Arkowitz: But the definition of Wikipedia here https://en.wikipedia.org/wiki/Cone_(topology)#:~:...
4
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1answer
59 views

Fibrantly generated model category

An important concept in the study of model categories is that of "cofibrantly generated model categories". These are nice because all morphisms can be obtained from a small subset of them ...
3
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1answer
48 views

Are discrete fibrations closed under pullback?

Let $P:E\to B$ be a discrete fibration of categories. Let $F:A\to B$ be a functor. Is the (strict) pullback $F^*P:A\times_B E \to A$ still a discrete fibration? Also, is there a reference where this ...
3
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2answers
54 views

Self homeomorphism of the cube

Let $I^n$ be the $n$-cube $[0,1]^n$. Also define two subsets of $\partial I^n$: $A=\{(x_1,\ldots,x_n)\mid x_1=0\}$ $B=\partial I^n\setminus \{(x_1,\ldots,x_n)\mid x_1=1\}$ So $A$ is the "bottom ...
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1answer
43 views

Fibration in integral manifolds

Consider a smooth manifold $M$ of dimension $n$ and an integrable tangent distribution $$ \mathcal{D} = span\{X_1,...,X_k\}$$ with $k\leq n$. Then we know that $M$ is foliated by the connected ...
2
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1answer
60 views

Pulback of a map to a pushout / pullback of cells in a fibration

Given a fibration $f:X \to B$ of CW complexes, it makes sense to guess that the pullbacks of a cell of $B$ will be a cell for $X$. That is, let $B_p$ be the $p$-th skeleton of $B$ and $X_p$ the ...
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0answers
46 views

Is there an isomorphism in long exact sequence of fibration, which arises from homotopy fiber construction?

I have inclusion of topological spaces $f:A\to B$. Then there is a fibration $E_f \xrightarrow{} B$, where $E_f$ stands for mapping path space. As I understand, $A$ is homotopy equivalent to $E_f$. ...
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1answer
50 views

Long exact sequence of the Klein bottle as a $S^1$-fiber bundle

If we look at he Klein bottle $K$ as a $S^1$-fiber bundle over $S^1$, we can apply the long exact sequence in Homotopy for fibers. $$\pi_2(S^1)\rightarrow\pi_2(K)\rightarrow\pi_2(S^1)\rightarrow\pi_1(...
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1answer
25 views

For $p: (E, e_0) \to (B, b_0)$ a Hurewicz fibration, is the inclusion $p^{-1}(b_0) \to E$ a Hurewicz fibration as well?

As $p$ is a Hurewicz fibration, its homotopy fiber $hofib_{b_0}(p)$ is homotopy equivalent to $p^{-1}(b_0)$, but does this also mean that the map $p^{-1}(b_0) \to E$ is a Hurewicz fibration as well?
3
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1answer
68 views

Show that $S^2$ and $S^3 \times \mathbb CP^\infty$ have isomorphic homotopy groups.

I want to show that $S^2$ and $S^3 \times \mathbb CP^\infty$ have isomorphic homotopy groups in each degree. My first approach was to calculate the homotopy group of $\mathbb CP^\infty$ and use the ...

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