projective bundle

(see also *Chern-Weil theory*, parameterized homotopy theory)

The concept of *projective bundle* is the generalization of that of *projective space* from vector spaces to vector bundles.

For $\pi \colon E \to X$ a topological vector bundle over some topological field $k$, write $E \setminus X \subset E$ for its complement of the zero section, regarded with its subspace topology.

Then its *projective bundle* is the fiber bundle $P(E) \to X$ whose total space the quotient topological space of $E \setminus X$ by the equivalence relation

$\left(
v_1 \sim v_2
\right)
\;\Leftrightarrow\;
\left(
(\pi(v_1) = \pi(v_2))
\,\text{and}\,
\left(
\underset{c \in k \setminus \{0\}}{\exists}
( v_2 = c v_1)
\right)
\right)$

hence

$P(E) \coloneqq (E \setminus )/\sim
\,,$

and whose bundle projection is

$\array{
P(E) &\longrightarrow& X
\\
[v] &\mapsto& \pi(v)
}
\,.$

- Klaus Wirthmüller, section 5 of
*Vector bundles and K-theory*, 2012 (pdf)

Last revised on April 3, 2019 at 06:53:39. See the history of this page for a list of all contributions to it.