nLab basic complex line bundle on the 2-sphere

Idea

The basic line bundle on the 2-sphere is the complex line bundle on the 2-sphere whose first Chern class is a generator $\pm 1 \in \mathbb{Z} \,\simeq\, H^2(S^2, \mathbb{Z})$, equivalently the tautological line bundle on the Riemann sphere regarded as complex projective 1-space.

This is the pullback bundle of the map $S^2 \to B U(1) \simeq B^2 \mathbb{Z}$ to the classifying space/Eilenberg-MacLane space which itself represents a generator of the homotopy group $\pi_2(S^2) \simeq \mathbb{Z}$.

Beware that this basic line bundle is sometimes called the “canonical line bundle on the 2-sphere”, but it is not isomorphic to what in complex geometry is called the canonical bundle of the 2-sphere regarded as a Riemann surface. Instead it is “one half” of the latter, its theta characteristic. See also at geometric quantization of the 2-sphere.

The basic line bundle is the canonically associated bundle to basic circle principal bundle: the complex Hopf fibration. Another name for it is the tautological line bundle for the complex projective line $\mathbb{P}^1(\mathbb{C})$ (the Riemann sphere), namely the map $\mathbb{C}^2 \setminus \{(0, 0)\} \to \mathbb{P}^1(\mathbb{C})$ mapping $(x, y)$ to $[x; y]$.

Definition

The classifying space for circle principal bundles, or equivalently (via forming associated bundles) that of complex line bundles is $B U(1)$, which as a Grassmannian is the infinite complex projective space $\mathbb{C}P^\infty$. The homotopy type of this space is that of the Eilenberg-MacLane space $K(\mathbb{Z},2)$. This means that $K(\mathbb{Z},2)$ is in particular path-connected and has second homotopy group the integers: $\pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z}$.

It being the classifying space for complex line bundles means that

$$\left\{ \array{ \text{isomorphism classes of} \\ \text{complex line bundles} \\ \text{on}\,\, S^2 } \right\} \;\simeq\; \left\{ \array{ \text{continuous functions} \\ S^2 \longrightarrow K(\mathbb{Z},2) \\ \text{up to homotopy} } \right\} \;\simeq\; \pi_2(K(\mathbb{Z},2)) \;\simeq\; \mathbb{Z} \,.$$

The (isomorphism class) of the complex line bundle which corresponds to $+1 \in \mathbb{Z}$ under this sequence of isomorphisms is called the basic complex line bundle on the 2-sphere.

Hence the basic complex line bundle on the 2-sphere is the pullback bundle of the universal complex line bundle on $B U(1)$ along the map $S^2 \to B U(1)$ which represents the element $1 \in \mathbb{Z} \simeq \pi_2(B U(1))$. If the classifying space $B U(1)$ is represented by the infinite complex projective space $\mathbb{C}P^\infty$ with its canonical CW-complex structure (this prop.), then this map is represented by the canonical cell inclusion $S^2 \hookrightarrow\mathbb{C}P^\infty$.

Notice that there is a non-trivial automorphism of $\mathbb{Z}$ as an abelian group given by $n \mapsto -n$. This means that there is an ambiguity in the definition of the basic line bundle on the 2-sphere.

Properties

Lemma

(clutching construction of the basic line bundle)

Under the clutching construction of vector bundles on the 2-sphere, the basic complex line bundle on the 2-sphere is given by the transition function

$$\mathbb{C} \supset \, S^1 \longrightarrow GL(1,\mathbb{C}) \, \subset \mathbb{C}$$

from the Euclidean circle $S^1 \subset \mathbb{R}^2 \simeq \mathbb{C}$ to the complex general linear group in 1-dimension, which is $GL(1,\mathbb{C}) \simeq \mathbb{C} \setminus \{0\}$ given simply by

$$z \mapsto z \,,$$

Alternatively, due to the sign ambiguity in the definition of the basic bundle, its clutching transition function is given by

$$z \mapsto - z \,.$$

Proof

Under the clutching construction the isomorphism class of a complex line bundle corresponds to the homotopy class of its clutching transition function

$$S^1 \to GL(1, \mathbb{C}) \simeq \mathbb{C} \setminus \{0\}$$

hence to an element of the fundamental group $\pi_1(\mathbb{C} \setminus \{0\}) \simeq \mathbb{Z}$. Hence by definition, the basic bundle has clutching transition function corresponding to $\pm 1 \in \mathbb{Z} \simeq [S^1, GL(1,\mathbb{Z})]$ and this element is represented by the function $z \mapsto \pm z$.

Proposition

(fundamental tensor/sum relation of the basic complex line bundle)

Under direct sum of vector bundles $\oplus_{S^2}$ and tensor product of vector bundles $\otimes_{S^2}$, the basic line bundle on the 2-sphere $H \to S^2$ satisfies the following relation

$$H \oplus_{S^2} H \;\simeq\; \left( H \otimes_{S^2} H \right) \oplus_{S^2} 1_{S^2}$$

(where $1_{S^2}$ denotes the trivial vector bundle complex line bundle on the 2-sphere).

(e.g (Hatcher, Example 1.13))

Proof

Via the clutching construction there is a single transition function of the form

$$S^1 \longrightarrow GL(n,\mathbb{C})$$

that characterizes all the bundles involved. With $S^1 \hookrightarrow \mathbb{C}$ identified with the topological subspace of complex numbers of unit absolute value, the standard choice for these functions is

• for the trivial line bundle $1_{S^2}$ we may choose

  $f_1 \colon z \mapsto \left( 1 \right)$;

• for the basic line bundle we may choose (by lemma )

  $f_H \colon z \mapsto \left( z\right)$

This yields

• for $H \otimes H \oplus 1_{S^2}$ the clutching function

  $z \mapsto \left( \array{ z^2 & 0 \\ 0 & 1 }\right)$

• for $H \oplus H$ the clutching function

  $z \mapsto \left( \array{ z & 0 \\ 0 & z } \right)$.

Since the complex general linear group $Gl(n,\mathbb{C})$ is path-connected (by this prop.), there exists a continuous function

$$\gamma \colon [0,1] \longrightarrow GL(2,\mathbb{C})$$

connecting the identity matrix on $\mathbb{C}^2$ with the one that swaps the two entries, i.e.

with $\gamma(0) = \left( \array{ 1 & 0 \\ 0 & 1 } \right)$

and $\gamma(1) = \left( \array{ 0 & 1 \\ 1 & 0 } \right)$

Therefore the function

$$\array{ S^1 \times [0,1] &\overset{}{\longrightarrow}& GL(2,\mathbb{C}) \\ (z,t) &\overset{\phantom{AA}}{\longrightarrow}& f_{H \oplus 1}(z) \cdot \gamma(t) \cdot f_{1 \oplus H}(z) \cdot \gamma(t) }$$

(with matrix multiplication on the right) is a left homotopy from $f_{H \oplus H}$ to $f_{H \otimes H \oplus 1}$.

Remark

(fundamental product theorem in topological K-theory)

Under the map

$$Vect(S^2)_{/\sim} \longrightarrow K(X)$$

that sends complex vector bundles to their class in the topological K-theory ring $K(X)$, the fundamental tensor/sum relation of prop. says that the K-theory class $H$ of the basic line bundle in $K(X)$ satisfies the relation

$$\begin{aligned} (H - 1)^2 & = H^2 + 1 - \underset{= H^2 + 1}{\underbrace{2 H}} \\ = & 0 \end{aligned}$$

in $K(X)$.

(Notice that $H-1$ is the image of $[H]$ in the reduced K-theory $\tilde K(X)$ of $S^2$ under the splitting $K(X) \simeq \tilde K(X) \oplus \mathbb{Z}$ (by this prop.).)

It follows that there is a ring homomorphism of the form

$$\array{ \mathbb{Z}[h]/\left( (h-1)^2 \right) &\overset{}{\longrightarrow}& K(S^2) \\ h &\overset{\phantom{AAA}}{\mapsto}& H }$$

from the polynomial ring in one abstract generator, quotiented by this relation, to the topological K-theory ring.

It turns out that this homomorphism is in fact an isomorphism, hence that the relation $(H-1)^2 = 0$ from prop. is the only relation satisfied by the basic complex line bundle in topological K-theory.

More generally, for $X$ a topological space, then there is a composite ring homomorphism

$$\array{ K(X) \otimes \mathbb{Z}[h]/((h-1)^2) & \longrightarrow & K(X) \times K(S^2) & \longrightarrow & K(X \times S^2) \\ (E, h) &\overset{\phantom{AAA} }{\mapsto}& (E,H) &\overset{\phantom{AAA}}{\mapsto}& (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H) }$$

to the topological K-theory ring of the product topological space $X \times S^2$, where the second map is the external tensor product of vector bundles.

This composite is an isomorphism if $X$ is a compact Hausdorff space (for $X = \ast$ the point space this reduces to the previous statement).

This is called the fundamental product theorem in topological K-theory. It is the main ingredient in the proof of Bott periodicity in complex topological K-theory.

Related concepts

• universal complex line bundle

• tautological line bundle, equivariant tautological line bundle

References

• Allen Hatcher, Vector bundles and K-theory (web)

• Klaus Wirthmüller, Vector bundles and K-theory, 2012 (pdf)