About Homology

  • #1
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.
 

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  • #2
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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.
 
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  • #3
Infrared
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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.
 
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  • #4
WWGD
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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.
 

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