# Tagged Questions

A fiber bundle is the most general kind of bundle. Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., vector bundles and principal bundles.

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### Are submersions of differentiable manifolds flat morphisms?

Let $\pi \colon M\to N$ be a smooth map between real smooth manifolds. Then $C^\infty(M)$ forms a module over $C^\infty(N)$ (via pullback). Is this module flat when $\pi$ is a submersion? Recall that ...
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### Looking for an introduction to orbifolds

Is there any source where the basic facts about orbifolds are written and proved in full detail? I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more ...
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### When are fiber bundles reversible?

My question, in its most general form is this: Given a fiber bundle $F\rightarrow E\rightarrow B$, when is there a fiber bundle $B\rightarrow E\rightarrow F$? Here, F,E, and B can lie in ...
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### Characteristic classes of sphere bundles over spheres in terms of clutching functions

I'm trying to understand Milnor's proof of the existence of exotic 7-spheres. Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...
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### Cohomology of fibrations over the circle: how to compute the ring structure?

This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
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### Which principlal bundles are locally trivial?

If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial. In the wikipedia article on ...
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### When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$

For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$ In my case ...
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### What manifold has $\mathbb{H}P^{odd}$ as a boundary?

This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting. ...
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### What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
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### Cohomology classes annihilated by pullbacks

A friend of mine is interested in examples of the following situation: an oriented smooth fiber bundle $\pi \colon M \to B$ with $M$ and $B$ compact and a non-zero class $a \in H^3(B; \mathbb{Q})$ ...
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### When do two holonomy maps determine flat bundles that are isomorphic as just bundles (w/o regard to the flat connections)?

Suppose we have a surface S (although the question might make as much sense in higher dimensions) and a topological group G. The data of a flat vector bundle on S (up to isomorphism) is the same as a ...
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### In how many ways can an iterated tangent bundle (T^k)M be viewed as a fibre bundle over (T^(k-1))M?

Let M be a smooth manifold. The double tangent bundle, TTM,can be viewed as a fibre bundle over TM in two ways, with the projection maps given by T_πM (i.e. the derivative of the projection from TM ...
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### smooth sections of smooth fiber bundles

A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where $E,M$ are ...
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### Circle bundles over $RP^2$

Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified? One can determine the isomorphism classes of bundles using obstruction theory, but I am ...
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### representatives of the group of homotopy 7-spheres

In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere ...
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### Classification of holomorphic disc bundles

I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber bundle,...
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### Surface bundles over a surface

What can be used to distinguish two $\Sigma_g$-bundles over $\Sigma_h$ up to (1) homotopy? (2) homeomorphism? (3) fiberwise homeomorphism? (4) bundle isomorphism? And can these always be computed ...
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### Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles

Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle. Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...
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### A sphere bundle map

I think this may all be classical bundle-theory. But I'm trying to read some old papers on classifications of bundles and the following came up as questions I couldn't immediately answer: Consider ...
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### Two questions about sphere bundles

I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and ...
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### Stiefel-Whitney class of fibre bundles

Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...
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### Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$

It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact: If $G$ is a Lie group ...
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### Grassmannian bundle theorem

Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$. ...
I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist. Remarks: 1) A pullback in a certain category is defined as ...