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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# A cohomology class on surfaces

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