Lagrangian immersions and Bott periodicity (Q1091001)

For a given smooth complex p-plane bundle E over \(S^ n\) a Lagrangian immersion \(\hat{\i}_ E: W\to T^*{\mathbb{R}}^{m+1}\) is constructed, where \(m=2(n+p)+1\), W is the image of a diffeomorphism from \(S^{n+1}\times {\mathbb{R}}^{m-n}\). An element \(\gamma'(\hat{\i}_ E)\in \pi_{n+1}G\ell (m+1,{\mathbb{C}})\) is obtained by preceding the so-called Gauss homotopy class of î\({}_ E\) by the composite \(S^{n+1}\times \{0\}\to S^{n+1}\times {\mathbb{R}}^{m-n}\cong W\). The main result gives an interpretation of \(\gamma'(\hat{\i}_ E)\) in terms of the Bott map. As a corollary, the author proves that as the bundle E varies the invariant \(\gamma'(\hat{\i}_ E)\) takes on all values in \(\pi_{n+1}G\ell (m+1,{\mathbb{C}})\).