I What are the applications of homology in understanding spaces and shapes?

[trees and plants]

Hello there.This is about homology.In homology as I know we also study holes in spaces, so a circle has a hole, so does a sphere but quite differently.If we have half a circle or a somehow not quite closed curve but almost closed curve could we study it with groups or something like homology?Thank you.

 

[fresh_42]

You would examine it with the methods which represent the structural aspects you are interested in:

• topology
• algebraic topology
• analysis
• geometry
• differential geometry

and additional methods like group actions dependent on these approaches. There is no answer until you settled what this curve is to you:

• a one dimensional space
• a connected, retractable space
• a graph of a function
• a segment of a circle
• a curved space

The properties you are interested in determine the method, not the object itself.

 

[Infrared]

If you have a singular chain $\sigma\in C_n(X)$, not necessarily closed (as your "half circle" isn't), but its boundary is a chain in a subspace $A\subset X$, then you can view $\sigma$ as representing a class in the relative homology group $H_n(X,A).$

There is a similar notion for relative homotopy groups, where elements of $\pi_n(X,A)$ are (based) homotopy classes of maps $(D^n,\partial D^n)\to (X,A)$ such that $\partial D^n$ is mapped to $A$ throughout the homotopy.

 

[WWGD]

It comes down ultimately to defining your cycle group, boundary group of the space at each "level" /dimension. That given, the factor n-groups will define the nth homology. An open arc as I understood you meant is contractible so all its homology groups are trivial. Harder, I would think is to study spaces like Gl(n, R), speces of continuous maps, etc., without any obvious geometry that I can tell.