Incompressibility and global inversion (Q1959004)

The main purpose of the paper is to use geometric methods in order to obtain new analytic conditions under which one can detect global injectivity and global invertibility. The main result is: Let \(n\geq 2\), \(f= (f_1,f_2,\dots, f_n): \mathbb{R}^n\to\mathbb{R}^n\) be \(C^1\) local diffeomorphism, \(1\leq k\leq n\) and \(H\) an affine subspace of codimension \(k\). Assume that \[ \int^\infty_0 \underset{\| x\|= r}{}{\text{inf}}\;{\|\bigwedge_{1\leq j\leq n}\nabla f_j(x)\|\over\|\bigwedge_{1\leq j\leq n,j\neq i}\nabla f_j(x)\|}\,dr= \infty \] for each \(i=1,\dots, k\). Then \(f^{-1}(H)\) is non-empty and \(\pi_j(f^{-1}(H))= 0\), \(j= 0,1,\dots, n- k\). In particular, \(f^{-1}(H)\) is non-empty and connected.