Questions tagged [fibration]

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

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4
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1answer
27 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 ...
2
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1answer
37 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 ...
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2answers
47 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
28 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
52 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
40 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
30 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
22 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
41 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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0answers
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Hitchin fibration and twisted connections

Let $p$ be a positive prime. Let $X,Y$ be smooth projective curves over a scheme $S$ of characteristic $p$, and let $\pi:X\times_S Y \to X$ be the first projection. Let $\mathcal E$ be a vector ...
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1answer
38 views

Closed inclusion

I have a simple question in the context of (co)fibrations in the context of Model Categories: Why on the page $52$ in the snippet below $$g^{-1}(d)$$ must be a single point not in the image of $A$ ? ...
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1answer
47 views

Fibrations are thought of as epimorphisms

In the book More concise algebraic topology on the page 213 they write We think of fibrations as analogous to epimorphisms. BUT Hovey on the page 51 says $f$ is a fibration if it is in $J-inj$. My ...
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0answers
20 views

Is it ever useful to consider cofibrations which satisfy the HEP only with respect to a proper subclass of spaces?

A map $j:A\rightarrow X$ is said to have the homotopy extension property with respect to a space $Z$ if whenever given a map $f:X\rightarrow Z$ and a homotopy $H:A\times I\rightarrow Z$ with $H_0=fj$, ...
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0answers
29 views

Explicit expression for cohomology transgression

I need help with my homework problem. For a fibration $$F \to \mathcal{E} \xrightarrow{p} B$$ prove that the transgression $\tau: E^{0, m-1}_m \to E^{m, 0}_m$ coincides with the composition $$E^{...
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1answer
46 views

How can an inclusion of finite groups induce a fibration of classifying spaces?

Let $G$ be a compact Lie group and $H$ be a closed subgroup. The inclusion $H \rightarrow G$ induces a homotopy fibration $G/H \rightarrow BH \rightarrow BG$. In particular, this must hold if $G$ and ...
2
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1answer
38 views

What does Hatcher mean by “pullback fibration” of a characterstic map?

I'm looking at Hatcher's chapter on spectral sequences and can't tease out the meaning of a statement early in the proof of the existence of the Serre spectral sequence (on homology). The goal at this ...
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2answers
210 views

Show that the orthogonal group acts transitively on the sphere $S^n.$

Show that the orthogonal group $$ O(n + 1) = \{ A \in GL(n+1 , \mathbb{R}) \mid A^{-1} = A^{T}\}$$acts transitively on the sphere $S^n,$ with stabilizer subgroup $O(n).$ Then use this to determine, ...
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1answer
80 views

Show that the space $Y = S^3 \vee S^6$ has precisely two distinct homotopy classes of comultiplications.

Here is the question: A comultiplication for a pointed space $X$ is a map $\phi : X \rightarrow X \vee X$ so that the composite $$X \xrightarrow{\phi} X \vee X \xrightarrow{i_{X}} X \times X$$ is ...
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1answer
49 views

(c) Calculate the homotopy fibre of the inclusion $i_{X} : X \vee X \rightarrow X \times X. $

Here is the question: Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Here is the answer of this part: Show that $\pi_{i}(...
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29 views

Identify the homotopy type of $F.$

Here is the question: Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Here is the answer of this part: Show that $\pi_{i}(...
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1answer
46 views

Show that $\pi_{i}(F) \cong \pi_{i +1}(X) $ where $F$ is the homotopy fiber of the inclusion $X \rightarrow X \times X.$

Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Could anyone explain to me how to prove this, please?
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1answer
25 views

Pullback of cartesian fibration preserves cartesian morphisms?

Let $$P:X \rightarrow B$$ be a cartesian fibration of ordinary one category. Let $F:C \rightarrow B$ be an arbitrary functor. Let $$Y \xrightarrow{F^*P} C$$ be the pullback of $P$ along $F$. Then it ...
4
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1answer
101 views

Show that either $X$ or $Z$ is homotopy equivalent to a point.

Prove or disprove the following statement: Suppose $X,Y,$ and $Z$ are simply connected $CW$ complexes and that $X \rightarrow Y \rightarrow Z$ is simultaneously a cofiber sequence and a fiber ...
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1answer
35 views

Spectral sequence with field coefficients

In the situation of the Serre spectral sequence for a fibration $F \rightarrow E \rightarrow B$, when can I say that the cohomology of $E$ with coefficients in a field is the direct sum of the ...
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0answers
22 views

Do locally trivial and smooth fibrations correspond to subbundles of the tangent bundle

Let $M$ be a manifold of dimension $n$. Let $p:E\rightarrow M$ be a locally trivial smooth fibration. Does this give us a way to construct a subbundle of the tangent bundle $TE$ of the manifold $E$, ...
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1answer
56 views

What is the fiber of the path space fibration?

Schematically, I understand the path space fibration $PX$ over some path-connected, pointed topological space $X$ with base point $x_o$ as: $$\Omega X \hookrightarrow PX \twoheadrightarrow X,$$ where ...
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0answers
80 views

How can the monodromy of an elliptic fibration have 0's on its diagonal?

This is maybe a silly question (but hopefully not). I am trying to understand the local monodromy of elliptic fibrations, and I am running into a seeming contradiction I do not know how to resolve. ...
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1answer
37 views

What does “generic fibre” mean in elliptic fibrations?

I've just started to read about elliptic surfaces in algebraic geometry. Here's a quote from wikipedia: An elliptic surface is a surface that has an elliptic fibration [...] such that almost all ...
2
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1answer
40 views

Pullback against a fibration invariant under homotopic maps

So I am currently trying to piece together little bits of knowledge I have acquired about fibrations in various context when I came across this question. If $p : E \rightarrow B$ is a Serre fibration ...
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0answers
18 views

Differential geometry: vector bundle cover of base that consist of charts that are also trvialising neighbourhoods [duplicate]

I am taking a first course in differential geometry, and we have looked at vector bundles and principle G bundles (I suppose that the following will hold for fibre bundles in general where the base is ...
2
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1answer
84 views

A small question about fiber bundle [duplicate]

I'm recently studying fiber bundles,and I find many examples of them are motivated from maps like $p:E\to B$ with homeomorphic fiber $F=p^{-1}(b)$.Therefore I'm wonder if the converse is true,that is ...
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0answers
38 views

Under what conditions can one deloop the free loop fibration?

Inspired by the MO question Homotopy extension of $E_\infty$-spaces. Sending a map $f:S^1\to X$ to $f(\text{basepoint})$ gives a fibration $\Lambda X\to X$ with fiber $\Omega X$, where $\Lambda X$ is ...
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1answer
39 views

Horizontal forms on principal $\mathbb{S}^1-$bundle

Assume that $M$ is a principal $\mathbb{S}^1-$bundle, i.e. an $\mathbb{S}^1-$manifold with the property that the action $M\times \mathbb{S}^1\to M$ is free (no fixed points) and the orbits $\mathcal{O}...
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1answer
45 views

Show that a path $\beta: I \rightarrow B $ lifts to a path $\bar{\beta}: I \rightarrow E .$

The question is: Let $p: E \rightarrow B $ be a fibration. Show that a path $\beta: I \rightarrow B $ with $\beta(0) = b_{0}$ and $e_{0} \in p^{-1}(b_{0})$ lifts to a path $\bar{\beta}: I \...
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1answer
33 views

The monodromy of a Lefschetz fibration as right-handed Dehn twists

A fact that one can find in many books is that the monodromy of a Lefschetz fibration is the product of right-handed/positive Dehn twists (one for each vanishing cycle). The only proof I could find ...
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0answers
46 views

The existence of Hom fibration between families of coherent sheaves

Let $S$, $X$ be algebraic varieties, and suppose $X$ is a protective smooth curve. And let both $\mathcal{G}$, $\mathcal{G}'$ be families of coherent sheaves on $X$ parameterized by $S$, i.e. $\...
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1answer
106 views

Show that $\phi$ is a fibration.

Show that if is a pullback square and $p$ is a fibration, then $\phi$ is a fibration. I know the definition of a pullback square and the definition of a fibration, but still, I do not know how ...
2
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1answer
64 views

Fibration induced by cofibration: Still surjective?

On "Concised Course of Algebraic Topology", a fibration is defined to be surjective. And there is a proposition: If $i:A\to X$ is a cofibration and $B$ is a space, then the induced map $p=B^i:B^X\...
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1answer
133 views

Using Hopf fibration to calculate $\pi_{3} (S^2)$

The question says: Theorems of Hurewicz and Hopf say that for $k < n, \pi_{k}(S^n)=1$ and $\pi_{n}(S^n)\cong \mathbb{Z}$. Assuming this for the moment, use the Hopf fibration $\eta : S^3 \...
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0answers
55 views

Understanding thm. 4.65 in AT.

The theorem and its proof are given below: 1-But I do not understand the proof very well, could someone give me a source for a more detailed proof, please? 2-My professor said at the end of the ...
3
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3answers
150 views

Basic obstruction theory : where does the obstruction to uniqueness of lifting lie?

This is a question about a remark someone said to me without giving much precision. Suppose you have two nice spaces $X,Y$ and are trying to build a map $X\to Y$ with certain nice properties. ...
3
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0answers
153 views

Is any smooth fibre bundle a smooth Hurewicz fibration?

From https://pdfs.semanticscholar.org/e737/a4f8b93242910c050c2faf761236dcf60f64.pdf (Theorem 2.1) it follows: (*) If $\pi:P\rightarrow M$ is a topological fibre bundle over a paracompat hausdorff ...
4
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1answer
116 views

Details in May's proof that local fibrations are fibrations

J. P. May proves on pages 49–50 (51–52 in the online pdf) of A Concise Course in Algebraic Topology that local (Hurewicz) fibrations are (Hurewicz) fibrations: Theorem. Let $p: E \to B$ be a map ...
2
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1answer
88 views

Homotopy equivalence of fibres of a Hurewicz fibration

Let $p:E \rightarrow B$ be a Hurewicz fibration. It is known that if $B$ is path connected, then the fibres over any two points in $B$ are homotopy equivalent. Question: Is there a simple proof of ...
2
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3answers
131 views

Are the $\Bbb S^2\times \Bbb R^2$ and $\Bbb R^2\times \Bbb S^2$ homeomorphic?

Are the $\Bbb S^2\times \Bbb R^2$ and $\Bbb R^2\times \Bbb S^2$ homeomorphic? I know that the answer is certainly yes but what is confused me is the following: $\Bbb R^2\times \Bbb S^2$: Consider a ...
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1answer
36 views

Weakly contractible space and extension

I wonder why in this proof: since the fibre $p^{-1}(b_0)$ is weakly contractible. Hence $h′_{\alpha}$ extends to a map $h_{\alpha}:D^n\to E$? I want to show if the boundary (the sphere of radius 1/2) ...
2
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1answer
52 views

Acyclic fibration admits a section?

Is that true that each acyclic fibration admits a section? If so how to see this? If not then under what condition an acyclic fibration admits a section?
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1answer
40 views

Example of Symplectic fibration

Can you give a example of manifold $M$ such that $\pi:M\longrightarrow T^2 $ be a Symplectic fibration?
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70 views

Fiber bundle and local trivial fibration

If we have a fiber bundle $\pi:M\longrightarrow N$ which M and N be two compact smooth manifolds,is there a fibration on $M$? If we have a surjective submersion $\pi:M\longrightarrow N$ such that ...
2
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1answer
128 views

A local trivial fibration

If M is compact and connected manifold. Let $\pi:M\longrightarrow T^{2} $ be a local trivial fibration. Why $\pi^{*}:H^{1}(T^{2})\longrightarrow H^{1}(M)$ is injective?

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