# Tagged Questions

For questions on vector bundles.

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### Normal bundle of a point

Let $X$ be a projective variety over a field $k$. I am trying to understand the notion of the normal bundle of a closed immersion. Let $x$ be a closed point of $X$. What is the normal bundle of $x$ ...
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### Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$

On page $10$ of Hatcher's Vector Bundles and K Theory, he gives a proof that the Whitney sum of the trivial line bundle over $\mathbb{R}P^n$ and the tangent bundle is equal to the Whitney sum of ...
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### What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
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### parallelizable sphere product closed disk

From Wall's Surgery on Compact Manifolds, P9: Observe that $S^r \times D^{m−r}$ is parallelisable. If $m > r$, this is true, because spheres can be embedded in Euclidean space of one ...
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### Chern classes of a double cover

Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ ...
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### Are there characteristic classes for symplectic vector bundles?

Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ...
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### Whitney sum bundle vs. direct product bundle [closed]

The question is simple: are Whitney sum bundle and direct product bundle the same? When are they different? PS: I have realized my mistake: let $E_1 \to B$ and $E_2 \to B$ be two bundles, their ...
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### What does “factoring out an (group) action $\tau$ of a group $G$ acting on some set $E$” mean?

I am reading a survey article where they define the following objects: $\Gamma:=\mathbb{Z}^{n}$ seen as a group of translations. $\mathbb{T}:=\mathbb{R}^{n}/\Gamma$ is the $n$-dimensional ...
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### Isomorphic modules of sections imply isomorphic bundles

For reference this is about a part of question 3-F in Characteristic Classes by Milnor and Stasheff, which discusses the $C(B)$-module structure of the space of sections of a topological vector bundle ...
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### Short Exact Sequence of Vector Bundles

Just wish to clarify, is it true that in order to show some vector bundles (over the same space) fit into a short exact sequence we just need to check that their fibers fit into a short exact sequence ...
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### On the zero in the fibre of a vector bundle.

Let $X$ be a differentiable manifold, let $\{U_i \mid i\in I\}$ be an open cover of $X$, let $\{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\}$ be a set of differentiable maps satisfying the cocycle condition, ...
I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...