Vector Bundles on P1

1 Topological Vector Bundles

Topologically, \(\PP ^1\) is \(S^2\). Since \(S^2\) is a suspension, complex vector bundles on \(S^2\) are given by \(\pi _2(BGL_n(\CC ))=\pi _1(GL_n(\CC ))=\ZZ \). The last equality can be interpreted as looking at how the determinant of a matrix varies in a loop. In particular, one finds that \(\cO (1)\) is the generator when \(n=1\), and adding copies of the trivial bundle gives the generators for \(n>1\). Using our classification we get the relation \(\cO (n)\oplus \cO (m)=\cO (n')\oplus \cO (m')\) whenever \(n+m=n'+m'\). Then one finds that the semiring of vector bundles is \(\NN [\gamma ,\gamma ^{-1}]/\gamma ^n+\gamma ^m=\gamma ^{n'}+\gamma ^{m'}\) for \(n+m=n'+m'\), where \(\gamma \) is \(\cO (1)\). Taking the Grothendieck group, this simplifies to \(K_0(\PP ^1)= \ZZ [\gamma ]/(\gamma -1)^2\).

As a remark, we have done this calculation very sloppily. If you are really careful, the identification \(K_0(\PP ^1)= \ZZ [\gamma ]/(\gamma -1)^2\) is so canonical that it can be done in families i.e. for a line bundle \(\xi \) over a space \(X\) one can show \(K_0(\PP (\xi +1)) = K_0(X)[\gamma ]/(\gamma -1)(\gamma \xi -1)\) (see Atiyah’s K-Theory book). When \(\xi \) is trivial, this gives a Kunneth formula in K-theory that gives Bott periodicity, with \(\gamma -1\) as the Bott class.