# Questions tagged [fibration]

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

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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...