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Results tagged with vector-bundles
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user 318
A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.
23
votes
Accepted
Are homology spheres stably parallelisable?
Yes, they have stably trivial tangent bundles. A remark to this effect can be found on page 70 of
M. Kervaire "Smooth Homology Spheres and their Fundamental Groups"
but it is a little terse. It is e …
- 18.2k
4
votes
Accepted
triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle
Yes, you can conclude that, because the construction
$$(\wedge^2 \xi \setminus \text{zero section})/\mathbb{R}_{>0} \to X/(\mathbb{Z}/2)$$
recovers the original double cover.
- 18.2k
3
votes
Accepted
A question on parallelizability
No: any finite set $S \subset M$ can be contained in the interior of an embedded closed disc in $M$, and cutting this out gives a manifold diffeomorphic to $M - \{x\}$. So if $M - S$ were parallelisab …
- 18.2k
15
votes
Accepted
Vector bundle for prescribed Stiefel-Whitney classes
No, because the Wu formulae express $\mathrm{Sq}^j(w_i)$ in terms of $w_k$'s, so if the $x_i$ you choose don't satisfy this formula, they cannot possibly arise as Stiefel--Whitney classes.
- 18.2k
5
votes
Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?
No: for example, there is no 1-manifold with Stiefel--Whitney number for $w_1$ equal to 1.
- 18.2k