A cohomology class on surfaces

In summary, the conversation is discussing a mod 2 cohomology class on a compact surface and its geometric meaning. The participants suggest triangulating the surface and taking the first barycentric subdivision to define a new triangulation. A 1-cochain is then defined on this new triangulation, which is a mod 2 cocycle. It is noted that this may be the first Stiefel-Whitney class of the surface, but it is later determined to be a coboundary.
  • #1
lavinia
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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.
 
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  • #2
never mind. This cohomology class is a coboundary.
 

1. What is a cohomology class?

A cohomology class is a mathematical concept used in topology and algebraic geometry to study the properties of spaces. In simple terms, it is a way of assigning algebraic objects to the points of a space that captures information about its structure and shape.

2. How is a cohomology class defined on a surface?

On a surface, a cohomology class is defined as a collection of closed forms (differential forms that have zero exterior derivative) that are representative of a certain cohomology group. These forms can be thought of as assigning a value to every point on the surface, much like a function.

3. What is the significance of studying cohomology classes on surfaces?

Studying cohomology classes on surfaces allows us to understand and classify different types of surfaces. It also helps us to distinguish between surfaces that are topologically equivalent, but have different geometric properties. This has important applications in fields such as physics and engineering.

4. How are cohomology classes computed on surfaces?

To compute cohomology classes on surfaces, we use techniques such as de Rham cohomology or sheaf cohomology. These involve representing the surface as a collection of smaller, simpler pieces, and then studying the relationships between these pieces to determine the cohomology class.

5. What are some real-world examples of cohomology classes on surfaces?

In physics, cohomology classes on surfaces are used to study the properties of electromagnetic fields and their interactions with surfaces. In engineering, they are used to analyze the stress and strain distributions on curved surfaces. They also have applications in computer graphics and animation, where they are used to model and manipulate the shapes of 3D surfaces.

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