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

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Fibered product of Hopf Fibrations

I am wondering: what is the vector bundle associated to the Hopf-fibration $S^{3}\to S^{2}$? What is the fibered product of two Hopf-fibrations? Thanks.
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8 views

About fibrations with isomorphic smooth fibers

Let $S$ be a smooth projective (complex) surface with a fibration $f:S\longrightarrow B$ over a base curve $B$. If all the fibers of $f$ are isomorphic to $F$, which is smooth, can I conclude that ...
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1answer
70 views

Rational map on $\mathbb P^1$ and its fibers

Consider a non-singular complex projective surface $S$ and a rational map $\psi:S\longrightarrow \mathbb P^1$; moreover suppose that $\psi$ is not defined on $\Delta=\{x_1,\ldots,x_m\}\subset S$. Now ...
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0answers
20 views

Two different notions of covering homotopy?

In Steenrod's The Topology of Fibre Bundles, there are two different notions of covering homotopy. One is furnished in Theorem 11.3: Let $\mathcal B,\mathcal B'$ be two bundles having the same ...
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27 views

Serre fibration and CW-complexes

Suppose that $p:X\rightarrow E$ is a Serre fibration. I know the definition: then $p$ has the right lifting property with respect to all inclusions $I^n\rightarrow I^{n}\times I$. Now it seems that ...
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1answer
41 views

Points with smooth fibers form an open subset

Let $S$ be a complex non-singular projective surface and let $C$ be a complex non-singular projective curve. Moreover consider a morphism $\varphi:S\longrightarrow C$ which is flat, proper and with ...
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23 views

Fibration with CW-complex as basespace admits retraction

Suppose $f:E\rightarrow B$ has the right lifting property with respect to all CW-pairs $(X,A)$. Then $f$ is a Serre fibration and also a weak-homotopy equivalence. But want i want to prove is the ...
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1answer
44 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 $G$. ...
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36 views

Sequence of cofibrations gives strict commutativity of given triangle

The problem is the following: Suppose $X_0\rightarrow X_1\rightarrow\cdots$ is a sequence of cofibrations. We will denote $f_n:X_n\rightarrow X_{n+1}$. Assume that we have also maps ...
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1answer
42 views

Fiber bundles and manifolds

Each vector bundle is an example of a fibre bundle with some extra structure. This extra structure provides the algebraic object consisting of all sections (continuos or smooth) of given bundle. When ...
2
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1answer
65 views

Relative $CW$-complexes and Serre fibrations

Suppose we have a (continuous) map $p:E\rightarrow X$. Assume that $p$ has the right lifting property with respect to any relative $CW$-complex $i:A\rightarrow B$. I want to show that $p$ is a Serre ...
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2answers
43 views

Fibration $p_1:P(Y,y_0)\to Y$ has section iff $Y$ is contractible.

Let $P(Y,y_0) = \{ \omega : \omega(0) = y_0 \}$ be path space let's consider a fibration $p_1:P(Y,y_0)\to Y$ such that $\omega \mapsto \omega(1)$. Show that there exists $s: Y \to P(Y,y_0)$ such ...
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1answer
49 views

fibration of real projective space over complex projective space

From the fibration $$U(1)=S^1\to S^{2n+1}\to \mathbb{C}P^n,$$ can we quotient the action of $S^0$ and obtain a well-defined fibration $$ S^1/S^0=\mathbb{R}P^1\cong S^1\to ...
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1answer
42 views

fibration of complex projective space over quaternion projective space

From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
3
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1answer
57 views

Postnikov towers for non-CW spaces

In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary. Question: Does ...
3
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3answers
128 views

$p:E\to B$ is fibration then $p_*:map(X,E)\to map(X,B)$ is fibration as well.

$p:E\to B$ is fibration then for $X$ being compactly generated weakly Hausdorff space $p_*:map(X,E)\to map(X,B)$ is fibration as well. We'd like to show that for any $Y$ and continuous $f$ and ...
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26 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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29 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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0answers
47 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 ...
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1answer
94 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 ...
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1answer
95 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?
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1answer
91 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
46 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 ...
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1answer
55 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 ...
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1answer
35 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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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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66 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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78 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 ...
2
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1answer
39 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 ...
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2answers
85 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 ...
3
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1answer
113 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 ...
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1answer
45 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 ...
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1answer
67 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
136 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
75 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
51 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 ...
2
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1answer
75 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 ...
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1answer
48 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
48 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 ...
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1answer
88 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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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 ...
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1answer
63 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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491 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 ...
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1answer
269 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: $$BH \to BG \to BK.$$ ...
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1answer
281 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 $ ...
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1answer
89 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
1k 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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348 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 $r \circ i = id$?
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2answers
200 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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1answer
192 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, ...