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basic complex line bundle on the 2-sphere

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Bundles

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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 of H 2(S 2,)H^2(S^2, \mathbb{Z}).

This is the pullback bundle of the map S 2BU(1)B 2S^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 π 2(S 2)\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 1()\mathbb{P}^1(\mathbb{C}) (the Riemann sphere), namely the map 2{(0,0)} 1()\mathbb{C}^2 \setminus \{(0, 0)\} \to \mathbb{P}^1(\mathbb{C}) mapping (x,y)(x, y) to [x;y][x; y].

Definition

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

It being the classifying space for complex line bundles means that

{isomorphism classes of complex line bundles onS 2}{continuous functions S 2K(,2) up to homotopy}π 2(K(,2)). \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+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 BU(1)B U(1) along the map S 2BU(1)S^2 \to B U(1) which represents the element 1π 2(BU(1))1 \in \mathbb{Z} \simeq \pi_2(B U(1)). If the classifying space BU(1)B U(1) is represented by the infintie complex projective space P \mathbb{C}P^\infty with its canonical CW-complex structure (this prop.), then this map is represented by the canonical cell incusion S 2P S^2 \hookrightarrow\mathbb{C}P^\infty.

Notice that there is a non-trivial automorphism of \mathbb{Z} as an abelian group given by nnn \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

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

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

zz, z \mapsto z \,,

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

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

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

Proposition

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

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

H S 2H(H S 2H) S 21 S 2 H \oplus_{S^2} H \;\simeq\; \left( H \otimes_{S^2} H \right) \oplus_{S^2} 1_{S^2}

(where 1 S 21_{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 1GL(n,) S^1 \longrightarrow GL(n,\mathbb{C})

that characterizes all the bundles involved. With S 1S^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 21_{S^2} we may choose

    f 1:z(1)f_1 \colon z \mapsto \left( 1 \right);

  • for the basic line bundle we may choose (by lemma 1)

    f H:z(z)f_H \colon z \mapsto \left( z\right)

This yields

  • for HH1 S 2H \otimes H \oplus 1_{S^2} the clutching function

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

  • for HHH \oplus H the clutching function

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

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

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

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

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

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

Therefore the function

S 1×[0,1] GL(2,) (z,t) AA f H1(z)γ(t)f 1H(z)γ(t) \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 HHf_{H \oplus H} to f HH1f_{H \otimes H \oplus 1}.

Remark

(fundamental product theorem in topological K-theory)

Under the map

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

that sends complex vector bundles to their class in the topological K-theory ring K(X)K(X), the fundamental tensor/sum relation of prop. 1 says that the K-theory class HH of the basic line bundle in K(X)K(X) satisfies the relation

(H1) 2 =H 2+12H=H 2+1 = 0 \begin{aligned} (H - 1)^2 & = H^2 + 1 - \underset{= H^2 + 1}{\underbrace{2 H}} \\ = & 0 \end{aligned}

in K(X)K(X).

(Notice that H1H-1 is the image of [H][H] in the reduced K-theory K˜(X)\tilde K(X) of S 2S^2 under the splitting K(X)K˜(X)K(X) \simeq \tilde K(X) \oplus \mathbb{Z} (by this prop.).)

It follows that there is a ring homomorphism of the form

[h]/((h1) 2) K(S 2) h AAA H \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 (H1) 2=0(H-1)^2 = 0 from prop. 1 is the only relation satisfied by the basic complex line bundle in topological K-theory.

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

K(X)[h]/((h1) 2) K(X)×K(S 2) K(X×S 2) (E,h) AAA (E,H) AAA (π X *E)(π S 2 *H) \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 XX \times S^^X is a compact Hausdorff space (for X=*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.

References

Revised on June 21, 2017 07:46:15 by Urs Schreiber (131.220.184.222)