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Results tagged with vector-bundles
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user 65800
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.
5
votes
1
answer
329
views
characteristic classes of tangent bundle of 2-nd unordered configuration space
Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\ …
- 2,223
6
votes
1
answer
295
views
Stiefel-Whitney class of unordered configuration space
Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-Whit …
- 2,223
5
votes
2
answers
796
views
triviality of Whitney sums of a vector bundle
Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid\sum_{i=1}^4x_i=0\}.$$ The alternating group $A_4$ acts on $V$ by
$$\sigma(x_1,x_2,x_3,x_4)=(x_{\sigm …
- 2,223
1
vote
1
answer
285
views
triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle
Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle
$$
\xi: \ma …
- 2,223
3
votes
0
answers
125
views
group actions of $S^3$ on the configuration space of projective plane
Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P …
- 2,223