Algebraic Topology – Difference Between Cellular, Simplicial, and Singular Homology

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I am confused about different types of complexes in algebraic topology. I am reading and using terms from Allen Hatcher´s Algebraic Topology. My confusion comes from how more things is called "simplex" or "complex".

Could you please check if I understand this correctly? I kind of know what is the difference between simplicial and singular homology, but I dont know where cellular homology stands. I also dont know how are the complexes and simplices different for these homologies. As far as I know, singular homology uses maps from standard simplices to the space, so it can be built "weirdly", while simplicial homology requires the space to be described by "nice" simplices. But I am not sure.

Thank you.

  • Simplicial complex – used for what? It is a bunch of n-simplices on the given space. An n-simplex where lengths of edges are $1$ is called a standard n-simplex. We have to keep track of the ordering of the vertices?

  • $\Delta$-complex – used for simplicial homology. Also consisting of n-simplices but somehow different? (E.g. They dont have to be determined by their vertices and different faces of simplices can coincide.)

  • CW complex – used for homotopy and for cellular homology? Is defined inductively by attaching n-cells and glueing them along their boundary.

  • Singular complex – used for singular homology. Consists of free abelian groups generated by the sets of n-simplices in the space. But n-simplices in this context are maps from standard n-simplices $\Delta^n$ to the space.

To my understanding, cellular homology is more general than simplicial (in the sense that simplicial homology cannot be always applied, e.g. for manifolds that cannot be triangularized).

However, if cellular homology works with simplicial complexes and simplicial homology with $\Delta$-complexes, I have read that $\Delta$-complexes are a generalization of simplicial complexes.

How come that the more general concept (cellular homology) is built using something less general (simplicial complexes)?

Best Answer

Let's zoom out a bit. A definition of homology has to navigate a tradeoff between several different nice properties it could satisfy, most notably a tradeoff between

  1. how easy it is to compute in examples, and
  2. how easy it is to prove theorems about.

Singular homology is the easiest homology to prove theorems about; it takes as input a topological space $X$ on the nose, no need for any kind of cell decomposition of $X$ or whatever, so it is obviously homeomorphism invariant, and even obviously functorial with respect to arbitrary continuous maps. Plus it is by far the easiest definition of homology to prove homotopy invariance for; this makes it a good "base" definition of homology to which others can be compared. Unfortunately because singular chain groups are so incredibly large, singular homology is very inconvenient for doing computations with directly. You basically only ever use singular homology to prove theorems.

Simplicial and cellular homology are both optimized for being easy to compute; you can get a hint of this from the fact that they have much smaller chain groups, typically finite-dimensional. It is extremely non-obvious that they are homotopy invariant, since they depend on so much additional structure (a triangulation and a cellular decomposition respectively); a priori it's not even obvious that they are homeomorphism invariant. It is also quite annoying to make either of these constructions functorial with respect to continuous maps. You can prove all this by proving that they agree with singular homology, or proving simplicial approximation resp. cellular approximation.

I recommend completely ignoring the concept of a $\Delta$-complex; as far as I know Hatcher is the only one who uses it. I think they were designed to be intermediate between simplicial complexes and CW complexes but in practice what everyone uses is CW complexes anyway. When most people talk about "simplicial homology" they are referring to simplicial complexes. (You are absolutely correct that it is very annoying to keep track of all the different things that are called a "simplex" or "simplicial" or a "complex" here. Patience! Learn one thing at a time!)

To get a concrete sense of the difference between all three of these definitions I suggest trying to write down all the chain groups and differentials for the $2$-sphere $S^2$. For singular homology you'll get some huge unmanageable infinite-dimensional thing. For simplicial homology you need to triangulate the $2$-sphere. And for cellular homology you need to find a cellular decomposition of the $2$-sphere; pay attention in particular to the difference between these things (you need more simplices for a triangulation than you need cells for a cell decomposition). Then you can try using each of these chain complexes to compute the homology of $S^2$.

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