Is This Mod 2 Cohomology Class on Surfaces Geometrically Significant?

[lavinia]

I am wondering if the following mod 2 cohomology class which can be defined on any compact surface, has any geometric meaning or is important in any way.

triangulate the surface then take the first barycentric subdivision. This is a new triangulation.

Define a 1 - cochain on this new triangulation as 1 on any 1- simplex that touches the barycenter of one of the 2 - simplices in the original triangulation and zero on any other 1 simplex. This is a mod 2 cocycle which is easily seen by drawing a picture.

I wonder if this is the first Stiefel-Whitney class of the surface.

 

[lavinia]

never mind. This cohomology class is a coboundary.