Homotopy classification of strict contact immersions
Let \(\Phi\) denote the sheaf of solutions of some \(r\)-th order partial differential relation \(R\) \(J^r(M,N)\) defined for \(C^r\) maps \(M\to N\), and \(\Psi\) the sheaf of sections of the \(r\)-jet bundle \(J^r(M,N) \to M\) with images in \(R\). The solution sheaf \(\Phi\) and the relation \(R\) are said to satisfy the parametric \(h\)-principle if the \(r\)-jet map \(j^r:\Phi\to\Psi\) is a weak homotopy equivalence. \(R\) is said to satisfy the \(h\)-principle if given a section of \(R\) one can deform it to a solution of \(R\) in \(\Psi\). The space of all strict contact immersions \((M, \eta) \to(N, \lambda)\) with \(C^\infty\) compact-open topology is denoted by \(\text{Cont}_s (M,N) \). Here \(\lambda\) is a 1-form on \(N\), while \(\eta\) is a subbundle of \(TM\) of codimension \(\leq 1\). Then the main result of the paper is: Theorem 1. The differential \(d:\text{Cont}_s (M,N) \to C_s (TM,TN)\) is a weak homotopy equivalence. In particular, strict contact immersions satisfy the parametric \(h\)-principle. Theorem 2. Contact immersions \((M,\eta) \to(N, \lambda)\) satisfy the ordinary \(C^0\)-dense \(h\)-principle.
scientific article; zbMATH DE number 1038398