A counting formula for the Kervaire semi-characteristic
Let \(M\) be a \((4q+1)\)-dimensional smooth, oriented, closed manifold. The Kervaire semi-characteristic \(k(M)\) is the mod 2 sum of the dimensions of the even-dimensional de Rham cohomology groups. The author proves a counting formula for this topological invariant, defined as follows. Let \(E\) be a codimension-one, orientable subbundle of the tangent bundle \(TM\) (which always exists since the Euler characteristic of \(M\) is zero), and let \(X\) be a section of \(E\) whose image is transverse to the zero section. Then the zero set of \(X\) consists of a finite union of circles. The behavior of \(X\) near each circle canonically determines a line bundle over that circle; the quantity \(k(M)\) is the (mod 2) number of these circles whose associated line bundles are trivial. To prove this, the author expresses the invariant as the mod 2 index of a family of elliptic operators on even forms on \(M\). The result is obtained by perturbing the family and localizing the calculation to the union of circles.
scientific article; zbMATH DE number 1436056