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

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916 views

An intuitive vision of fiber bundles

In my mind it is clear the formal definition of a fiber bundle but I can not have a geometric image of it. Roughly speaking, given three topological spaces $X, B, F$ with a continuous surjection $\pi: ...
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375 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
6
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2answers
718 views

Serre Spectral Sequence and Fundamental Group Action on Homology

I am looking at my algebraic topology notes right now, and I am looking at our definition for the Serre Spectral Sequence and it requires that the action of the fundamental group of the base space of ...
6
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1answer
57 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?
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641 views

Showing that the loopspace $\Omega S^{\infty}$ is homotopic to $S^{\infty}$.

Showing that the infinite dimensional sphere $S^{\infty}$ is contractible is rather easy by constructing an explicit contraction (Hatcher gives a nice one). I thought it might be a nice exercise to ...
4
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1answer
213 views

A short exact sequence of groups and their classifying spaces

Suppose that we have a short exact sequence of topological groups: $$1 \to H \to G \to K \to 1.$$ I have found some papers mentioning that the above sequence induces a fibration: $$BH \to BG \to BK.$$ ...
4
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1answer
123 views

Article or book explaining rigorously facts about the mapping class group

I would like to know more about relationships between the mapping class group of an orientable surface with negative Euler's characteristic and moduli spaces. In particular, I would like to have a ...
3
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2answers
183 views

Does $\Omega$ functor preserve fibration?

If $p:E \rightarrow B$ is a fibration, is it necessarily that $\Omega p:\Omega E \rightarrow \Omega B$ a fibration? (where $\Omega E$ is the loopspace of $E$ and so is $\Omega B$)
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2answers
260 views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
3
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1answer
101 views

Fibrations induced by deformation retractions

Given a map $f: A \to B$ and a homotopy equivalence $g: C \to A$, I wish to show that $E_f \to B$ and $E_{fg} \to B$ are fiber homotopy equivalent, where, given a function $f: A \to B$, $E_f$ is the ...
3
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1answer
180 views

Beck Chevalley condition and maps of adjunctions

Suppose we have a split fibration $p : \mathbb{E}\to\mathbb{B}$ with (split) simple products. To fix notation, this means that for every projection $\pi_{I,J} : I\times J \to I$ in the base category, ...
3
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1answer
224 views

3-manifolds fibering over the circle and mapping tori

If $S$ is a closed connected surface and $\varphi \in \mathrm{Diff}(M)$, then we can build the mapping torus $M_\varphi = \dfrac{S \times [0,1]}{(x,0)\sim (\varphi(x),1)}$. Then we have that $ ...
3
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0answers
340 views

Fibre and homotopy fibre

$p:E \rightarrow B$ is a fibration and $F$ is its fibre and $F_p$ its homotopy fibre. If $i:F \rightarrow F_p$ is the inclusion, is there a homotopy inverse $r$ of $i$ such that $r \circ i = id$?
2
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1answer
40 views

Fundamental crossed square of a square of spaces

I am a newbie at Topology and I didn't took the homotopy theory course. Sorry for that, but I need to know some facts about this paper ("Van Kampen Theorems for Diagrams of Spaces", authors R. Brown ...
2
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1answer
37 views

Proof that if $S^{k}\rightarrow S^{m}\rightarrow S^{n}$ is a fiber bundle, then $k=n-1$ and $m=2n-1$.

I was reading a proof online (7a) here: http://www.math.wisc.edu/~dummit/sets/752-fs.pdf that if $S^{k}\rightarrow S^{m}\rightarrow S^{n}$ is a fiber bundle, then $k=n-1$ and $m=2n-1$ but I'm ...
2
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2answers
412 views

Monomorphisms and fibrations are preserved by pullback

I just came across a strange property of morphisms that are preserved under pullbacks, and it made me wonder. Consider a model category $\mathcal{M}$. Because the fibrations are exactly the maps that ...
2
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1answer
114 views

Non-Kan Fibrations

In the definition of a Kan fibration (on nlab), i.e. for a map $\pi:Y\to X$ of simplicial sets the inclusion of any horn into $Y$ always lifts to an inclusion of the filled in horn if that filled in ...
2
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1answer
157 views

Composition of fibrations

Suppose that $p:E \to B$ and $q:B \to B'$ are fibrations. Is it true that $qp:E \to B'$ is a fibration? I thought this might just be 'abstract-nonsense'. There is a diagram (possibly ...
2
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1answer
63 views

What is a mapping class group, and how can we use it to understand fibrations of 3-manifolds when the fiber is a surface?

So I'm reading this book, kind of committed to reading the entire thing, and in the section I'm up to the author starts using some language regarding monodromies and mapping class groups. Having ...
2
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0answers
90 views

Every $\mathbb{P}^n$-bundle is a $\mathbb{P}(\mathscr{E})$

I am working on exercise II.7.10(c) in Hartshorne's Algebraic geometry, which asks: Let $X$ be a noetherian regular scheme. Show that every $\mathbb{P}^n$-bundle $P$ over $X$ is isomorphic to ...
2
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1answer
56 views

Graphic interpretation of path fibration.

Let $S^2$ the unit sphere. We can consider the associated path fibration $$ \Omega(S^2) \rightarrow P(S^2) \rightarrow S^2 .$$ I have to explain path fibration so I think that it is useful to make a ...
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2answers
79 views

Action of $SO_n$ on $\mathbb{S}^{n-1}$ induces fibre bundle.

Real compact Lie group $SO_n$ acts smoothly and transitively on $\mathbb{S}^{n-1} \subseteq \mathbb{R}^n$ with obvious action. Isotropy subgroup of each point in $\mathbb{S}^{n-1}$ is isomoprhic to ...
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1answer
62 views

Inverse limit of sequence of fibrations

I'm reading through the proof of Proposition 4.67 in Hatcher's Algebraic Topology, and I've come to something that I'm having trouble understanding. For an arbitrary sequence of fibrations $... ...
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1answer
62 views

Proof that Beck-Chevalley holds for right adjoints iff it holds for left adjoints

I am looking at Bart Jacob's book "Categorical Logic and Type Theory". The proof of Lemma 1.9.7 is left as an exercise for the reader. It does not seem that easy to me, and i have had quite limited ...
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1answer
80 views

Fibrations lift p-connectedness

Let $p: X \to B$ be a Serre fibration, and suppose that $B^p \subset B$ is a subspace of $B$ such that $(B,B^p)$ is $p$-connected, i.e. $\pi_n(B,B^p)=0 \ \forall n \leq p$ or, equivalently, ...
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1answer
43 views

Examples of non fibrations

What are some illustrating examples of functors $\mathcal{E} \to \mathcal{B}$ which are neither a fibration nor an opfibration? I've found many positive examples but I'm blanking out on negative ...
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1answer
78 views

Questions about fibers (pre-image)

Just want to check if I have done this correctly. Elegant, simple, concise answers are welcome and preferred. It took me a long time to get this far. Problem statement: Let $f: A → B$ be a function. ...
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1answer
57 views

Being contractible in homotopy theory vs. homotopy type theory

I'm trying to clarify the notion of being contractible in homotopy theory vs. homotopy type theory. Is the following right? "In homotopy theory the real interval $[0,1]$, considered as a subset ...
1
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1answer
29 views

Proving the left lifting property for a map

I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration $q:X \rightarrow Y$, where $q$ is a weak equivalence. I don't know how to draw a square ...
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2answers
245 views

How to prove that some mapping is a Serre fibration?

From Husemoller, page 8 (rephrased): Theorem: Let $E, B \in \mathbf{Top}_*$, and $F \to E \to B$ be a Serre fibration. Then there is a natural group homomorphism $\partial: \pi_n(B) \to \pi_{n-1}(F)$ ...
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15 views

When does a homogeneous space define a fibration?

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we ...
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28 views

Is $\operatorname{Fib}(B)$ cartesian closed when $B$ is?

Is the ($2$-)category $\operatorname{Fib}(B)$ of fibrations over $B$ cartesian closed whenever $B$ is? If not, are there some restrictions that would make it so? For example, consider restricting to ...
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58 views

morphisms of principal bundles with different structure groups

Let $f \,: X \to Y$ be a continuous map between spaces. Let $G$ and $H$ be topological groups. Consider the diagram: \begin{equation} \label{} \begin{array}{ccccccccccccccccccccccccccccccc} E_G ...
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66 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
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1answer
104 views

Understanding the Concept of Monodromy; case of Lefschetz Fibrations.

My question is on the concept of monodromy around critical points in a Lefschetz fibration $p: M^4 \rightarrow S^2$ (and monodromy in general), where $M^4$ is a 4-manifold and $S^2$ is the 2-sphere. ...
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1answer
78 views

When and why do vanishing cycles of Lefschetz fibrations exist?

On page 6 of the article Symplectic Lefschetz fibrations with arbitrary fundamental groups, the authors state that for a Lefschetz fibration (with total space of dimension 4) the retraction of the ...
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1answer
51 views

A problem concern about the relationship between homotopy groups.

I encountered the following problem: Let $X$ be a closed manifold, $S^n$ be the $n$-dimensional standard sphere and denote $\Omega(S^n,X)$ be the space of base point-preserving maps from $S^n$ to ...
0
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1answer
29 views

Regarding an arbitary fibration as an inclusion

I'm reading through Allen Hatcher's Algebraic Topology, and he mentions that, given a Postnikov tower, the fibration $X_n \rightarrow X_{n-1}$, where $X_n$ and $X_{n-1}$ are CW complexes, can be ...
0
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1answer
47 views

Classifying the different types of fibers on $V(uX^2 + vYZ) \subset \mathbb{P}^1 \times \mathbb{P}^2$

I'm working on an exercise where I'm supposed to note the different types of fibers on $V(uX^2 + vYZ) \subset \mathbb{P}^1 \times \mathbb{P}^2$. The way I was thinking to do this was to consider the ...
0
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1answer
49 views

M fibers over the circle then construct a symplectic form

I'm trying to prove that if a 3-manifold $M$ fibers over the circle, then $M\times S^1$ admits a symplectic structure. I know that it is an standard result. Probably it is very easy, but I can't see ...
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0answers
22 views

Ref. Request Lefschetz Fibrations, Restriction of Base

All. Let $ M^4 \rightarrow S^2 $ be a Lefschetz fibration over $S^2$, where $M^4$ is a compact, oriented 4-manifold. I am still weak in this topic, and I would appreciate references to properties ...
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41 views

Group action on fibre homotopy group

Let $p : E \rightarrow B$ be a Serre fibration, with typical fibre $F \cong p^{-1}(b)=: F_b$ for each $b \in B$. I know that $\pi_1(F) \looparrowright \pi_k(F)$, but now I would like to find a way to ...