The tag has no wiki summary.

learn more… | top users | synonyms

7
votes
0answers
229 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 ...
7
votes
1answer
687 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: ...
5
votes
2answers
600 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 ...
5
votes
3answers
600 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
votes
1answer
120 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
votes
2answers
206 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
votes
1answer
106 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: $$BH \to BG \to BK.$$ ...
3
votes
1answer
87 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
votes
1answer
145 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
votes
1answer
158 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
votes
0answers
329 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
votes
2answers
144 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 $\Omega B$)
2
votes
1answer
27 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
votes
2answers
320 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
votes
1answer
108 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
votes
0answers
50 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
votes
0answers
73 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
votes
1answer
55 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 ...
1
vote
2answers
58 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 ...
1
vote
1answer
49 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 $... ...
1
vote
1answer
52 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
66 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. ...
1
vote
2answers
212 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 $\partial: \pi_n(B) \to \pi_{n-1}(F)$ ...
1
vote
1answer
122 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 ...
1
vote
1answer
64 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. ...
1
vote
0answers
67 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 ...
0
votes
1answer
25 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
votes
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
votes
1answer
38 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 ...
0
votes
0answers
54 views

Fibre homotopy equivalence

$E \mathop \to \limits^p B$ and $E_1 \mathop \to \limits^{p_1} B$ are two fibrations and there is a map $f:E \rightarrow E_1$ such that $f$ is a homotopy equivalence and ${p_1} \circ f = p$. Are ...
0
votes
0answers
36 views

about principal fibration

Supposing $\Omega C\rightarrow E\rightarrow B$ is a principal fibration and has a section, is $E$ fiber homotopy equivalent to $\Omega C\times B$? This is Exercise 4.3.22 of Hatcher's book. Thanks.