Why Are Homology Groups Not MUCH Larger?

[Mandelbroth]

So far, I think algebraic topology is turning out to be the best thing since sliced bread. However, I'm having a bit of difficulty with homology, for one particular reason.

Consider, as an example, the first homology group of $S^1$. The definition of the free abelian group (or, in general, the $R$-module) of $n$-chains is the free abelian group generated by ALL $n$-simplices. Why do we not include ALL paths in $S^1$ in this definition? Are these not also 1-simplices?

Thank you!

 

[HallsofIvy]

You are not talking about "paths", you are talking about equivalence classes of paths. Many different paths lie in one equivalence class.

 

[Mandelbroth]

You are not talking about "paths", you are talking about equivalence classes of paths. Many different paths lie in one equivalence class.

AHA! Thank you! That makes soooooo much more sense. :rofl:

 

[WWGD]

To add a little bit, remember that homology is a quotient be it of modules or vector spaces --cycles/boundaries -- which shows the elements to be classes, where two cycles are equivalent (geometrically *) if their difference bounds . And in the case of $S^1$ , the classes are the paths that go around n times. You can see that if you go around a non-integer "number of times" k , then the path can be deformed/homotoped to the path class |_ k _|, where |_ k _| is the greatest integer less than k .* You can also do this purely algebraically.

 

[blue_raver22]

I can understand your confusion about why homology groups are not much larger. However, the reason for this is related to the fundamental concept of homology itself.

Homology is a mathematical tool used to study topological spaces, and it is based on the idea of "cycles" and "boundaries". In simple terms, a cycle is a continuous path or loop in a space that does not have any "holes" or "gaps" in it, while a boundary is a path that forms the boundary of a higher-dimensional object. The homology group of a space is the collection of all cycles modulo boundaries.

Now, coming to your example of the first homology group of $S^1$, the reason we do not include all paths in this definition is because not all paths in $S^1$ are considered as cycles. In fact, only closed paths (or loops) in $S^1$ are considered as cycles, while open paths are not. This is because open paths can be continuously deformed into each other without ever creating a "hole" or "gap", while closed paths cannot.

Therefore, the free abelian group generated by all 1-simplices in $S^1$ only includes the closed paths, as they are the ones that contribute to the homology group. Including all paths in the definition would result in a much larger homology group that does not accurately reflect the topological properties of the space.

In summary, homology groups are not much larger because they are based on the fundamental concept of cycles and boundaries, and not all paths in a space are considered as cycles. I hope this helps clarify your understanding of homology. Keep exploring algebraic topology, it truly is a fascinating field of study!