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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)?

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  • $\begingroup$ I really only understand simplical homology (and it is of doubt how much). But the idea is certain topological spaces can be triangulated (i.e: you can consider the space to be made of simplices). This is a topological isomorphism. After this, on the set of these simplifcal objects, you can talk about things like points, edges, faces etc. From these you can consider formal expressions (free group gererated by these objects) and defined a boundary oeprator which takes one from free groups of n dimensional object to n-1 dim. This boundary operator is a group isomorphism (contin) $\endgroup$ Commented Jul 26, 2022 at 23:57
  • $\begingroup$ it turns out some groups which can be constructured from considering the image and kernal of the boundary operator are topological invariant (sort of like how Euler's number is). $\endgroup$ Commented Jul 26, 2022 at 23:58
  • $\begingroup$ Oops: I meant boundary operator is homomorphism not isomorphism $\endgroup$ Commented Jul 27, 2022 at 0:06

4 Answers 4

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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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    $\begingroup$ @Tereza: yes, we can just take a point and a $2$-cell for the cellular decomposition, which involves no $1$-cells, so there's no differential; that gives a nice geometric picture of what the homology looks like. For the triangulation you need to find a way to build a $2$-sphere out of triangles (big hint: any convex polyhedron whose faces are triangles accomplishes this). Describing what singular homology is doing geometrically is very complicated because of how complicated an arbitrary continuous map from an $n$-simplex to a topological space can be; don't worry about this for now. $\endgroup$ Commented Jul 26, 2022 at 15:47
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    $\begingroup$ @Tereza: ah, I mean the maps in the definition of a chain complex, I guess Hatcher calls them boundary homomorphisms. $\endgroup$ Commented Jul 26, 2022 at 15:52
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    $\begingroup$ I also used to think that Hatcher just made up $\Delta$-complexes. But more recently I learned some of the deeper history of that concept, occurring in what are nowadays called simplicial sets, as explained here for instance. $\endgroup$
    – Lee Mosher
    Commented Jul 26, 2022 at 20:20
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    $\begingroup$ I also learned of $\Delta$-complexes under the name of semi-simplicial complexes. I do not recommend completely ignoring this concept, I find them very useful for computations for low-dimensional examples because semi-simplicial triangulations tend to be much smaller than simplicial ones. If I remember correctly, In my first algebraic topology course simplicial homology was defined for semi-simplicial complexes immediately. $\endgroup$ Commented Jul 27, 2022 at 9:00
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    $\begingroup$ @Tereza: simplicial homology can be defined using either simplicial complexes or $\Delta$-complexes; simplicial complexes are $\Delta$-complexes satisfying an additional condition so one is a special case of the other. $\endgroup$ Commented Jul 27, 2022 at 10:42
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This is just an extended comment to Qiaochu Yuan's perfect answer.

He writes that $\Delta$-complexes were designed to be intermediate between simplicial complexes and CW complexes. I do not think that is the essential point. Simplicial complexes are purely combinatorial objects whereas $\Delta$-complexes and CW complexes provide additional structures on topological spaces $X$ in form of a collection of maps $\sigma_\alpha : \Delta^n \to X$ which satisfy suitable requirements. For CW complexes one usually writes $\Phi_\alpha : D^n \to X$ (instead of $\sigma_\alpha : \Delta^n \to X$) and calls them characteristic maps. Topologically there is no difference between the standard $n$-simplex $\Delta^n$ and the $n$-ball $D^n$, but $\Delta^n$ has a combinatorial structure (it is the convex hull of the $n+1$ standard basis vectors $e_0,\ldots,e_n$ of $\mathbb R^{n+1}$ which have a natural ordering) giving us $n+1$ canonical face-embeddings $\iota_k : \Delta^{n-1} \to \Delta^n$. We can therefore say that each $\Delta$-complex is a CW complex with very special characteristic maps.

In my opinion $\Delta$-complexes are designed to provide a link between the singular chain complex and the cellular chain complex.

The singular chain complex $C_*(X) =(C_n(X),\partial)$ of a space $X$ is defined by taking $C_n(X)$ to be the free abelian group generated by all continuous maps $\sigma^n : \Delta^n \to X$. The boundaries $\partial : C_n(X) \to C_{n-1}(X)$ are explicitly defined via the face-embeddings $\iota_k : \Delta^{n-1} \to \Delta^n$.

For the cellular chain complex of a CW complex $X$ we take in dimension $n$ the free abelian group generated by all open $n$-cells of $X$. Unfortunately the boundaries are not that nice as for the singular complex; we need the Cellular Boundary Formula on p. 140. I think it may be very hard to explicitly determine the degrees $d_{\alpha \beta}$. Anyway, the benefit is that we may be able to make explicit computations of cellular homology groups. For singular homology groups we do not have a chance unless in very trivial cases; one usually takes an axiom-based approach for computations.

So what is the role of $\Delta$-complexes? For any $\Delta$-complex $X$ Hatcher defines a chain complex $\Delta_*(X) = (\Delta_n(X),\partial)$ by taking for $\Delta_n(X)$ the free abelian group generated by all open $n$-cells of $X$ (as in the celluar complex). In contrast the boundaries $\partial : \Delta_n(X) \to \Delta_{n-1}(X)$ are defined exactly as in the singular complex via the face-embeddings $\iota_k : \Delta^{n-1} \to \Delta^n$. This is what I called the link between the singular chain complex and the cellular chain complex. Note that in a $\Delta$-complex the open $n$-cells are in $1$-$1$-correspondence with the structure maps $\sigma_\alpha : \Delta^n \to X$. Thus we could alternatively define $\Delta_n(X)$ to be the the free abelian group generated by all these $\sigma_\alpha$. Doing so produces a genuine sub-chain-complex of the singular chain complex. One can say that we make a particular efficient selection of singular $n$-simplices which gives a drastically shrunken chain complex (which allows to make explicit computations of homology groups).

Update:

Hatcher introduces various types of objects to which he associates chain complexes (which then give us homology groups of these objects).

The singular chain complex is defined for all topological spaces by taking into account all continuous maps $\sigma : \Delta^n \to X$ which are called singular simplices. We do not need any additional structure on $X$, and that is the great benefit of this approach. The disadvantage is that the set of singular simplices is tremendously big so that the singular chain complex does not allow efficient computation of homology groups unless in very trivial cases. In other words, it is a very uneconomical approach.

As I explained above, $\Delta$-complexes and CW complexes can be regarded as topological spaces $X$ plus a certain collection of singular simplices in $X$. There is an intimate relationship between the topology of $X$ and this collection. For example, the images of the singular simplices must cover $X$ and $X$ must have the weak topology with respect to these images. But as you know there are more conditions.

CW complexes are a very interesting and commonly used class of objects for the purposes of algebraic topology. The $\Delta$-complexes form a subclass, but they do not seem to be popular in the literature. Their main benefit is that the "$\Delta$-chain-complex" is a genuine subcomplex of the singular chain complex. In contrast, for CW complexes the boundaries of the cellular chain complex are somewhat opaque.

What about simplicial complexes?

Hatcher introduces simplicial complexes, but does not define a simplicial chain complex (and thus no simplicial homology groups). Unfortunately he is a bit imprecise. A simplicial complex as defined on p.107 is a purely combinatorial object, but in the most parts of his book he has in mind what is usually called a geometric simplicial complex. A geometric $n$-simplex is the convex hull of $n+1$ points in general position in some Euclidean space and thus it is a topological space (in contrast to combinatorial, or abstract, simplices which are just finite sets). A geometric simplicial complex is a collection of such geometric simplices satisfying suitable conditions. Have a look for example in

  • Munkres, James R. Elements of algebraic topology. CRC Press, 2018

to get more information. Hatcher does not distinguish between "combinatorial" and "geometric" simplicial complexes. He does not even mention that there is a difference. On p.107 he says

From this combinatorial data a $\Delta$-complex $X$ can be constructed, once we choose a partial ordering of the vertices $X_0$ that restricts to a linear ordering on the vertices of each simplex in $X_n$. For example, we could just choose a linear ordering of all the vertices. This might perhaps involve invoking the Axiom of Choice for large vertex sets.

This is very sloppy. In fact, one can associate to each combinatorial simplicial complex a geometric simplicial complex (geometric realization), but Hatcher omits all details. See for example Munkres' book or

  • Spanier, Edwin H. Algebraic topology. Springer Science & Business Media, 1989.

To each geometric simplicial complex one can associate a $\Delta$-complex as described by Hatcher; it means that for each geometric simplex $s$ of the geometric simplicial complex we choose an affine isomorphism $\sigma : \Delta^n \to s$ such that suitable compatibility requirements are satisfied. This procedure is not unique since it is based on choices. In other words, geometric simplicial complexes are not a subclass of $\Delta$-complexes, the latter contain more infomation in form of the maps $\sigma : \Delta^n \to X$.

To each simplicial complex one can associate the simplicial chain complex. I shall not go into details, see Munkres or Spanier. This requires to introduce the concept of oriented simplices. Hatcher does not do that. Working with $\Delta$-complexes does not require this concept.

Summarizing, the essential difference between simplicial complexes and $\Delta$-complexes is that a simplicial complex $X$ is a collection of objects (simplices) which cover $X$, whereas a $\Delta$-complex $X$ is a collection of structure maps $\sigma : \Delta^n \to X$ whose images cover $X$.

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  • $\begingroup$ Thank you! However, I am maybe even more confused. In my question, there are 4 kinds of complexes. But there are 3 kinds of homologies mentioned. So I still dont get the difference between $\Delta$-complex and singular complex and singular complex. Which one is for simplicial homology and which one for singular? And are both three just "bunch of simplices or maps (in the case of singular complex) from standard simplices to the space"? Or are two of them name for the same thing? Or what am I missing? $\endgroup$ Commented Jul 27, 2022 at 10:35
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    $\begingroup$ As I said, my answer is just a sort of comment. If I find time, I shall make an update to my answer. Note that Hatcher introduces simplicial complexes, but does not define a simplicial chain complex (and thus no simplicial homology groups). As often, he is a bit imprecise. A simplicial complex as defined on p.107 is a purely combinatorial object, but you can associate to it a geometric realization which is a geometric simplicial complex. This is a space with an additional structure which is related to a $\Delta$-complex, but formally is no $\Delta$-complex. $\endgroup$
    – Paul Frost
    Commented Jul 27, 2022 at 23:10
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    $\begingroup$ Hatcher does not distinguish between "combinatorial" and "geometric" simplicial complexes. He does not even mention that there is a difference. $\endgroup$
    – Paul Frost
    Commented Jul 27, 2022 at 23:11
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Let me supplement the nice answer by Qiaochu Yuan: regarding simplicial complexes, they arise naturally in various circumstances. Abstractly, a simplicial complex is a set $X$ of finite sets, closed under taking subsets. (If $S$ is in $X$, so is any subset of $S$.) Given such a structure, its geometric realization is obtained by associating to a set $S \in X$ with cardinality $n+1$ an $n$-simplex. The elements of $S$ correspond to the vertices of that simplex, and any subset $T$ of $S$ corresponds to the face determined by the elements of $T$.

For example, there are lots of constructions in combinatorics and graph theory that produce simplicial complexes. See https://mathoverflow.net/questions/161586/what-have-simplicial-complexes-ever-done-for-graph-theory, for example.

The point is, simplicial complexes are natural mathematical structures, not just useful in topology as a way of computing homology groups.

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enter image description here

Page-21, Anatoly Fomenko, Visual Topology and Geometry

Clarifications:

  1. The invariance is of the homology groups under homotopy (I may have misunderstood the book here as it wasn't explicitly discussed)
  2. The book discusses the differences in details in the nearby pages. However, in the chapter the diagram is from (chap-1) , is on Simplicial Homologies.
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