7
votes
2answers
630 views

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 ...
9
votes
4answers
543 views

$S^n \to S^m \to B$ bundle: possible?

Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with. (1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the ...
8
votes
1answer
338 views

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 ...
7
votes
2answers
974 views

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 ...
6
votes
2answers
959 views

Totally geodesic surfaces in fibered 3-manifolds

Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface? (Of course such manifolds exist if the 'Virtually Fibered ...
10
votes
4answers
1k views

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 ...
17
votes
3answers
828 views

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 ...
2
votes
2answers
396 views

Branched coverings over orbifolds with reflector lines

It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via ...
2
votes
3answers
555 views

Reducible 3d torus bundles

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So, could anyone give me a hint to classify them? In contrast, do you agree ...