A note on \(\Gamma{}_ n^ C\)-structures (Q1181057)

Let \(\Gamma^{\mathbb{C}}_ n\) be the groupoid of germs of local analytical automorphisms of \(\mathbb{C}^ n\) and let \(B\Gamma^{\mathbb{C}}_ n\) be the classifying space for \(\Gamma^{\mathbb{C}}_ n\) structures. Denote by \(F\Gamma^{\mathbb{C}}_ n\) the homotopy fibre of the map \(B\Gamma^{\mathbb{C}}_ n\to BGL(n,\mathbb{C})\) naturally induced by the differential \(\Gamma^{\mathbb{C}}_ n\to GL(n,\mathbb{C})\). The main result of the article states that the homotopy group \(\pi_ i(F\Gamma^{\mathbb{C}}_ n)\) exactly describes the set of all integrable homotopy classes of \(PC\)-structures of type \((p,1)\) on the manifold \(S^ i\times \mathbb{R}^{m-i}\) provided \(n<i<2n\), \(m-i\geq 1\), \(m=2p+1\), \(n=p+1\). The concepts of a \(PC\)-structure, \(PC\)-foliation, \(PC\)-submersion, partial complex extension are briefly recalled; roughly they concern the theory of immersions of real manifolds into complex manifolds. The above main result is derived from a rather general \(PC\)-transversality theorem (which cannot be stated here).