Normally branched line fields on odd-dimensional manifolds
Let \(M\) be a manifold and let \(\Delta\subseteq M\) be a submanifold of codimension \(2\). Assume that \(\partial \Delta \subseteq \partial M\) and that \(\Delta\) is transverse to \(\partial M\). View the normal bundle of \(\Delta\) as a tubular neigborhood with associated spherical bundle \(M\odot \Delta\). Suppose we are given a branched line bundle \(L\to M\odot \Delta\) and a bundle embedding \(j: L_{\partial (M\odot \Delta)} \to TM|_{\partial (M\odot \Delta)}\) with only isolated nontrivial zeros (i.e., \(J\) is a regularly defect line field). When \(M\) is even-dimensional, a numerical invariant may be defined as follows: \( \widetilde \theta(M\odot \Delta, L, j) \in {\mathbb Z} \) is the sum of the indices of the zeros of \(v:L\to T(M\odot \Delta)\), where \(v\) is any extension of \(j\) which has only isolated zeros. A result of Jänich says that, when \(j\) is normally branched, there is a Poincaré-Hopf type formula relating \(\widetilde \theta (M\odot \Delta, L, j)\) to the Euler characteristics of the components of \(\Delta\) and the Frank indices of \(j\) along those components. For odd-dimensional manifolds, the indices are only defined modulo \(2\) and the resulting invariant is denoted by \(\theta(M\odot \Delta, L, j)\); this invariant does not satisfy such a simple formula in general. In this paper, \(\Delta\subseteq M\) with \(M\) even-dimensional as above, but the invariant \(\theta\) is studied for the data \(\partial \Delta\subseteq \partial M\) and the restrictions of \(L\) and \(j\). The main theorem includes the formula \[ \theta(\partial M\odot \partial \Delta, L|_{\partial M\odot \partial \Delta}, j_{\partial(\partial M\odot \partial \Delta)}) =\chi(\Delta)\quad \text{ mod} 2, \] thereby giving a simple formula in the special case where `the data bounds.'.
scientific article; zbMATH DE number 1901922