(see also Chern-Weil theory, parameterized homotopy theory)
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 $[x; y]$.
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
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 infintie complex projective space $\mathbb{C}P^\infty$ with its canonical CW-complex structure (this prop.), then this map is represented by the canonical cell incusion $S^2 \hookrightarrow\mathbb{C}P^\infty$.
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.
(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
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
Alternatively, due to the sign ambiguity in the definition of the basic bundle, its clutching transition function is given by
Under the clutching construction the isomorphism class of a complex line bundle corresponds to the homotopy class of its clutching transition function
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 $z \mapsto \pm z$.
(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
(where $1_{S^2}$ denotes the trivial vector bundle complex line bundle on the 2-sphere).
(e.g (Hatcher, Example 1.13))
Via the clutching construction there is a single transition function of the form
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
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
(with matrix multiplication on the right) is a left homotopy from $f_{H \oplus H}$ to $f_{H \otimes H \oplus 1}$.
(fundamental product theorem in topological K-theory)
Under the map
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
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
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
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.
Allen Hatcher, Vector bundles and K-theory (web)
Klaus Wirthmüller, Vector bundles and K-theory, 2012 (pdf)
Last revised on October 22, 2021 at 13:29:20. See the history of this page for a list of all contributions to it.