Is This Mod 2 Cohomology Class on Surfaces Geometrically Significant?

  • Context: Graduate 
  • Thread starter Thread starter lavinia
  • Start date Start date
  • Tags Tags
    Class Surfaces
Click For Summary
SUMMARY

The discussion centers on the geometric significance of a mod 2 cohomology class defined on compact surfaces. The process involves triangulating the surface and taking the first barycentric subdivision to create a new triangulation. A 1-cochain is defined on this triangulation, assigning a value of 1 to any 1-simplex touching the barycenter of a 2-simplex from the original triangulation, resulting in a mod 2 cocycle. Ultimately, it is concluded that this cohomology class is a coboundary, indicating it lacks independent geometric significance.

PREREQUISITES
  • Understanding of mod 2 cohomology
  • Familiarity with triangulation of surfaces
  • Knowledge of barycentric subdivision
  • Concept of coboundaries in cohomology theory
NEXT STEPS
  • Research the properties of Stiefel-Whitney classes
  • Explore the implications of coboundaries in algebraic topology
  • Study triangulation techniques for compact surfaces
  • Learn about mod 2 cohomology applications in geometry
USEFUL FOR

Mathematicians, topologists, and students interested in algebraic topology, particularly those studying cohomology classes and their geometric interpretations on surfaces.

lavinia
Science Advisor
Messages
3,385
Reaction score
760
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.
 
Physics news on Phys.org
never mind. This cohomology class is a coboundary.
 

Similar threads

Graduate Reference for simplicial homology and cohomology
  • · Replies 23 ·
Replies
23
Views
6K
Graduate Calculating the Z2 cohomology of the Klein Bottle using intersections
  • · Replies 3 ·
Replies
3
Views
6K
Graduate Proving that the real projective plane is not a boundary
  • · Replies 38 ·
2
Replies
38
Views
8K
  • Poll Poll
Geometry Algebraic Curves and Riemann Surfaces by Miranda
  • · Replies 1 ·
Replies
1
Views
5K
Graduate Deciding Between DG & PDE Graduate Math Classes
  • · Replies 14 ·
Replies
14
Views
4K
Surface roughness in 2D sinusoidal corrugation
  • · Replies 1 ·
Replies
1
Views
5K
Job Skills [Career] - What kind of life does a career in physics entail?
  • · Replies 2 ·
Replies
2
Views
4K
  • Poll Poll
Geometry A Comprehensive Introduction to Differential Geometry series by Spivak
  • · Replies 15 ·
Replies
15
Views
23K
Graduate BRS: Simplicial Homology with Macaulay2
  • · Replies 2 ·
Replies
2
Views
3K
Programs How good is this applied math program?
  • · Replies 13 ·
Replies
13
Views
4K