nLab projective bundle

Idea

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

Definition

For $\pi_E \colon E \to X$ 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 $P(\pi_E) \,\colon\, P(E) \to X$ whose total space is 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

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

and whose bundle projection is

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

Properties

Splitting

The pullback bundle of a vector bundle $E$ to its own projective bundle

$$\array{ P( \pi_E )^\ast E &\longrightarrow& E \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow {}^{\mathrlap{\pi}} \\ P(E) &\underset{P(\pi)}{\longrightarrow}& X }$$

naturally contains a line bundle, whose point-set definition is

$$\mathcal{L} \;\coloneqq\; \big\{ \left. ( \ell, v ) \,\in\, P(E) \times_X E \;\right\vert\; v \in \ell \big\} \;\subset\; P( \pi_E )^\ast E \,.$$

Hence the pullback splits as the direct sum of vector bundles of this $\mathcal{L}$ and its orthogonal complement (for any choice of inner product):

$$(P(\pi_E))^\ast (E) \;\simeq\; { \color{blue} \mathcal{L} } \oplus \big( P(\pi)^\ast(E) \big)/\mathcal{L} \,.$$

This process continues: Pulling back further to the projective bundle of $\mathcal{L}^{\perp}$ yields

$$(P(\pi_{\mathcal{L}^\perp}))^\ast (P(\pi_E))^\ast (E) \;\simeq\; { \color{blue} \mathcal{L}' } \oplus (P(\pi_{\mathcal{L}^\perp}))^\ast \big( {\color{blue} \mathcal{L} } \big) \oplus \big( (P(\pi_{\mathcal{L}^\perp}))^\ast P(\pi)^\ast(E) \big)/\big( P(\pi_{\mathcal{L}^\perp}))^\ast \mathcal{L} \oplus \mathcal{L}' \big) \,,$$

for some line bundle $\mathcal{L}'$

This way, eventually one finds a pullback bundle of $E$ which is entirely a direct sum of line bundles.

(e.g. Wirthmüller 12, p. 15 (17 of 67))

This is one incarnation of the splitting principle.

Examples

Complex-projective bundle of quaternionic tautological line bundle

Proposition

(complex projective bundle of quaternionic tautological line bundle is complex projective space)

For $n \in \mathbb{N}$, the projective bundle of the rank=2 complex vector bundle underlying the quaternionic (dual) tautological line bundle $\mathcal{L}_{{}_{\mathbb{H}P^n}}$ over quaternionic projective space $\mathbb{H}P^n$ is the complex projective space $\mathbb{C}P^{2n+1}$ equipped with the map that sends complex lines to the quaternionic lines that they span:

$$\array{ P_{\mathbb{C}} \big( \mathcal{L}^\ast_{{}_{\!\!\!\mathbb{H}P^n}} \big) && \simeq && \mathbb{C}P^{2n+1} \\ & \searrow && \swarrow_{ \mathrlap{ v \cdot \mathbb{C}^\times \,\mapsto\, v \cdot \mathbb{H}^\times } } \\ && \mathbb{H}P^n }$$

Proof

We compute as follows (showing this for the dual tautological bundle just for definiteness of notation):

$$\begin{aligned} P_{\mathbb{C}} \big( \mathcal{L}^\ast_{\!\!\!\mathbb{H}P^n} \big) & = \; \Big( \mathcal{L}_{{}_{\mathbb{H}P^n}} \setminus \big( \mathbb{H}P^n \times \{0\} \big) \Big) \big/ \mathbb{C}^\times \\ & = \; \Big( \big( \mathbb{H}^{n+1} \setminus \{0\} \big) \underset { \mathbb{H}^\times }{ \times } \big( \mathbb{H} \setminus \{0\} \big) \Big) \big/ \mathbb{C}^\times \\ & \simeq\; \Big( \big( \mathbb{H}^{n+1} \setminus \{0\} \big) \times \{1\} \Big) \big/ \mathbb{C}^\times \\ & \simeq \big( \mathbb{C}^{2n+2} \big) \big/ \mathbb{C}^\times \\ & \simeq \; \mathbb{C}P^{2n+1} \,. \end{aligned}$$

Here:

1. The first line is the definition of the complex projective bundle (here);

2. the second line inserts the definition of the tautological quaternionic line bundle (here);

3. the third line observes that, being away from its zero section, we have unique representatives of the elements in its defining quotient space whose fiber component is the unit $1 \in \mathbb{C} \subset \mathbb{H}$;

4. the fourth line identifies the evident underlying $\mathbb{C}^\times$-space;

5. the fifth line is the definition of complex projective space.

This shows that the total space is as claimed.

Moreover, since the projection map is the quotient space projection from this by the full $\mathbb{H}^\times$-action, the same computation with $(-)/\mathbb{C}^\times$ replaced by $(-)/\mathbb{H}^\times$ shows that the bundle map is as claimed.

Related entries

• Thom space

• fundamental product theorem in topological K-theory

• projective unitary group

• twisted K-theory, twisted equivariant K-theory

References

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