Enriched Euler numbers for line bundles

Project leader: Professor Kirsten Wickelgren
Project manager: Stephen McKean
Team members: John Igieobo, Steven Sanchez-Osorio, Dae'Shawn Taylor

Euler classes are used to help distinguish between line bundles over topological spaces and their algebraic analogs. Over the real numbers, the classical Euler class is computed by the Brouwer degree of a section “s”. If “s” intersects the zero section with positive slope, this intersection point contributes +1 to the Euler class; if “s” intersects the zero section with negative slope, this intersection point contributes -1 to the Euler class. Using this formula, one can show that the cylinder has Euler class 0, while the Möbius strip does not have Euler class 0. In particular, the cylinder and the Möbius strip are not the same line bundle.

For vector bundles over an algebraic scheme, there is a modified version of the Euler class, known as the A¹-Euler class, which is valued in bilinear forms. However, the A¹-Euler class is only defined in dimensions higher than 1.  We set out to define an unstable A¹-Euler class for line bundles over algebraic curves. In addition to the bilinear form given by the original A¹-Euler class, the unstable A¹-Euler class has an extra scalar value that can be likened to a determinant.