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

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698 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 ...
13
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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: ...
9
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1answer
149 views

Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
7
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2answers
998 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 ...
7
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1answer
164 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?
5
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1answer
378 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: $$BH \to BG \to BK.$$ ...
5
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3answers
739 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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2answers
217 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 $\Omega B$)
4
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1answer
68 views

Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
4
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1answer
70 views

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.
4
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1answer
132 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 ...
4
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0answers
40 views

Why generalized vectors can be written locally as sum of vectors and 1-forms?

I would like to understand better this point. In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle ...
4
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1answer
419 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
3answers
131 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 ...
3
votes
2answers
390 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
122 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
206 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
74 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
votes
1answer
71 views

examples for fibration not fibre bundle

We can use path space to make a map into a fibration. Generally, is this construction of fibration a fiber bundle? Or can someone give me some examples of fibration not fiber bundle?
3
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0answers
355 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
42 views

How to construct the horizontal bundle?

I am learning the concept of connection. I am confused by the construction of the horizontal bundle. My question is: For a fibre bundle $M\rightarrow B$, the vertical vector space $V$ can be easily ...
2
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1answer
66 views

question about “Homology fibrations and the group completion theorem”

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In ...
2
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1answer
98 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 ...
2
votes
2answers
72 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 ...
2
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1answer
159 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 ...
2
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1answer
55 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
108 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is ...
2
votes
1answer
83 views

Getting fiber bundles from short exact sequences

Are there conditions that guarantee that a split short exact sequence of groups $$ 1 \rightarrow K \rightarrow G \rightarrow Q \rightarrow 1 $$ gives rise to a fiber bundle $$ F \rightarrow E ...
2
votes
1answer
76 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 ...
2
votes
1answer
50 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
1answer
52 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 ...
2
votes
2answers
572 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
146 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
44 views

Given a column vector, can we add columns, continuously dependent on the given one, to get an invertible matrix?

Given a vector $x$ in the $n=6$-dimensional Euclidian space $\mathbb{R}^n$, do there exist $n-1$ continuous functions $f_1$ to $f_{n-1}$ such that the matrix $$(x,f_1(x), … ,f_{n-1}(x))$$ is ...
2
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0answers
18 views

Fibration induced by an almost complex structure

Let $E \rightarrow M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $$End(E)\rightarrow M,$$ denoted by ...
2
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0answers
87 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
2
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0answers
23 views

Polynomials as Locally Isotrivial Covers

Let $k=\mathbb{A}^1$ be algebraically closed of arbitrary characteristic. I am interested in understanding when a polynomial $f:\mathbb{A}^n\to\mathbb{A}^1$ defines a locally isotrivial family over ...
2
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0answers
30 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
2
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0answers
25 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 ...
2
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0answers
38 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 ...
2
votes
1answer
84 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
116 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
65 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
55 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
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1answer
78 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 ...
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2answers
90 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
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1answer
78 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
166 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
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1answer
82 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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1answer
97 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, ...