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Questions tagged [fibration]

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

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space (called a fiber) being "parameterized" by another topological space (called a base). A fibration is like a fiber bundle, except that the fibers need not be the same space, rather they are just homotopy equivalent. (Wikipedia)

Further reading : Fibre bundles

364 questions
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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 and bundle projection between them. Under some…
Lor
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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 $\pi: X\rightarrow B$, we "attach" to every point $b$…
Dubious
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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 space in a fiber bundle over $B$: namely…
rj7k8
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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 a fibration $F\to E\to B$ be trivial on all…
Jonathan Beardsley
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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: $$BH \to BG \to BK.$$ Here $B$ assigns to each (topological) group its…
H. Shindoh
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What is the essential difference between a sheaf and a fibration with lifting property?

Are there cases where the two notions coincide? By sheaf I mean a pre-sheaf ($\mathcal{C}^{op} \rightarrow \mathcal{Set}$) satisfying the sheaf condition. The sheaf condition says that, for every open cover of $U \subseteq X$ = opens sets of…
Yan King Yin
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interpreting a long exact sequence of homotopy groups

It is a well-known result in homotopy theory that a fibration $F \rightarrow E \rightarrow B$ induces a long exact sequence in the homotopy groups; namely, $$\pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F)\rightarrow…
C. Zhihao
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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 $ M_\varphi \to S^1$ is a fibration (it's actually a fiber…
Lor
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Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?
Jonathan Gleason
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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 homotopic to the diagonal map. Show that the space $X…
Intuition
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induced map in homology on a fiber bundle

Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map $f: E \rightarrow E$ that covers a map $g: B \rightarrow B$. If the maps $g_*: H_*(B, \mathbb{Q})…
C. Zhihao
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Understanding the Hopf Link

I am trying to understand why the preimages of two points under the Hopf fibration are linked. I thought that two circles in $\mathbb{C}^n$ are linked iff one circle intersects the convex hull of the other. $$p: S^3\to\mathbb{C}P^1,\quad…
SVG
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Example of a Serre fibration between manifolds which is not a fiber bundle?

I'm looking for an example of a map $f : X \to Y$, where $X$ and $Y$ are manifolds (without boundary), and $f$ is a Serre fibration, but $f$ is not a fiber bundle. I know that if $f$ is proper, and $f$ is a smooth fibration, then there are no such…
Elle Najt
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Why is $SU(n)/SU(n-1)$ the $2n-1$-sphere?

I am looking at Fomenko, Fuchs' book on "Homotopical Topology" and they claim that we have the isomorphism $$ SU(n)/SU(n-1) \cong S^{2n-1} $$ Why is this true? Here is what I have so far: If I have a matrix $A \in SU(n-1)$, then we can embed…
54321user
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Proof of the Group Completion Theorem [Hatcher], Fiberwise Suspension

I'm reading Hatcher's proof of the group completion theorem, in A short exposition of Madsen-Weiss Theorem, Pages 36 ~ 42. After Lemma D.3., Hatcher says that the total space is homotopy equivalent to $BM \vee BM$. I do not understand this. As far…
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