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Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old at the time) seemed to know what he was talking about and has had a not too unsuccessful career since.

So my (quite connected) questions are

1) Why do so few books use this approach ( I only know Massey's) ?

2) What are the pro's and con's of both approaches ?

3) Does it matter ? After all the homology groups obtained are the same.

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up vote 22 down vote accepted

Others have mentioned the advantages to cubical sets and so I don't want to say much on those; I'll just mention some facts about the other direction. The main disadvantage that comes solely from the point of view of homology is that you have this irritating normalization procedure: you have to take the cubical chains on X and take the quotient by degenerate cubes. (You can do the same for simplicial chains and you get the same answer as the original.) The cubical theory gets niceties that others have stated.

The main advantages of simplices are mostly apparent when you move a little further along into homotopy theory. The homotopy theory of simplicial sets is in some senses simply easier than the cubical analogue. For example, the cartesian product of two simplicial sets has as geometric realization the product of the geometric realizations of its factors. On the other hand, the standard "cubical interval" I, which realizes to [0,1], has a self-product I^2, whose geometric realization has fundamental group ℤ. Simplices in some ways play more nicely with degeneracies than cubes do.

Simplices are also more closely tied to categories via the simplicial nerve functor. For example, there are the simplicial constructions of classifying spaces coming from categories, and these give you nice constructions of group cohomology. (Perhaps I just don't know the cubical analogues.)

There is also the ubiquitous "bar construction" which is unreasonably useful in algebraic topology, and which comes most naturally from a simplicial construction. We use this to resolve modules, show equivalences between algebras over different operads, and more applications that are almost too numerous to name. For example, May used it quite heavily in his proofs that spaces X which are algebras over an E-operad have infinitely many deloopings BX, B2X, ...

There has been a lot of work done on cubical sets, however, but I'm not the best authority on it. Try here for a starting point.

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I think simplices are more convenient for constructing cones, while cubes are more convenient for constructing products. In fact, we can think of simplices as smallest collection of polyhedra which contains a point (0-simplex) and is closed under taking cones, while the cubes are the smallest collection which contain an interval (1-cube) and are closed under products. (Alternatively, contain a point and closed under taking product with an interval.)

Personally, I think it is more convenient to do singular homology with the larger collection of polyhedra which is closed under both cones and products. (The n-dimensional polyhedra in this collection are indexed by rooted trees with n edges. The simplices correspond to maximal depth trees where the valence of a vertex is at most 2, while the cubes correspond to minimal depth (star-shaped) trees where the root vertex has valence n and all the other vertices have valence 1.)

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Re: using singular homology using polyhedra obtained from both cones and products, that's a nice idea. Is it worked out anywhere? Does it admit a nice convenient notation? – Kevin H. Lin Nov 1 '09 at 15:05
I'm not aware of anywhere this has been written down, but that doesn't mean much since I'm not very well read. The underlying combinatorics is closely related to both dendroidal sets and blob homology. – Kevin Walker Nov 1 '09 at 15:34
People are using an approach, closed under both products and cones. The keyword to look for is prodsimplicial. – Mikael Vejdemo-Johansson Nov 1 '09 at 17:44
Cool, thanks. Weird sounding name though. Personally I think I'd go with "prisms". – Kevin H. Lin Nov 1 '09 at 17:55
There is a 1957 paper ``Supercomplexes'' by Victor Gugenheim that if I remember rightly goes towards combining simplices and cubes. – Peter May Dec 17 '11 at 3:16

Our new book (NAT)

Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15

uses mainly cubical, rather than simplicial, sets. The reasons are explained in the Introduction: in strict cubical higher categories we can easily express

algebraic inverse to subdivision,

a simple intuition which I have found difficult to express in simplicial terms. Thus cubes are useful for local-to-global problems. This intuition is crucial for our Higher Homotopy Seifert-van Kampen Theorem, which enables new calculations of some homotopy types, and suggests a new foundation for algebraic topology at the border between homotopy and homology.

Also cubes have a nice tensor product and this is crucial in the book for obtaining some homotopy classification results.

I have found that with cubes I have been able to conjecture and in the end prove theorems which have enabled new nonabelian calculations in homotopy theory, e.g. of second relative homotopy groups. So I have been happy to use cubes until someone comes up with something better. ($n$-simplicial methods, in conjunction with cubical ideas, turned out, however, to be necessary for proofs in the work with J.-L. Loday.)

See also some beamer presentations available on my preprint page.

Here is a further emphasis on the above point on algebraic structures: consider the following diagram:


From left to right pictures subdivision; from right to left pictures composition. The composition idea is well formulated in terms of double categories, and that idea is easily generalised to $n$-fold categories, and is expressed well in a cubical context. In that context one can conjecture, and eventually prove, higher dimensional Seifert-van Kampen Theorems, which allow new calculations in algebraic topology. Such multiple compositions are difficult to handle in globular or simplicial terms.

The further advantage of cubes, as mentioned in above answers, is that the formula $$I^m \times I^n \cong I^{m+n}$$ makes cubes very helpful in considering monoidal and monoidal closed structures. Most of the major results of the EMS book required cubical methods for their conjecture and proof. The main results of Chapter 15 of NAT have not been done simplicially. See for example Theorem 15.6.1, on a convenient dense subcategory closed under tensor product.

Sept 5, 2015: The paper by Vezzani arxiv::1405.4548 shows a use of cubical, rather than simplicial, methods, in motivic theory; while the paper by I. Patchkoria, HHA arXiv:1011.4870, Homology Homotopy Appl. Volume 14, Number 1 (2012), 133-158, gives a "Comparison of Cubical and Simplicial Derived Functors".

In all these cases the use of connections in cubical methods is crucial. There is more discussion on this mathoverflow. For us connections arose in order to define commutative cubes in higher cubical categories: compare this paper.

See also this 2014 presentation The intuition for cubical methods in algebraic topology

April 13, 2016. I should add some further information from Alberto Vezzani:

The cubical theory was better suited than the simplicial theory when dealing with (motives of) perfectoid spaces in characteristic 0. For example: degeneracy maps of the simplicial complex $\Delta$ in algebraic geometry are defined by sending one coordinate $x_i$ to the sum of two coordinates $y_j+y_{j+1}$. When one considers the perfectoid algebras obtained by taking all $p$-th roots of the coordinates, such maps are no longer defined, as $y_j+y_{j+1}$ doesn't have $p$-th roots in general. The cubical complex, on the contrary, is easily generalized to the perfectoid world.

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Ronnie Brown has for a long time been working with cubical strict higher categories and argues that, while for representing spaces the advantages of simplicial sets may be bigger, for carrying algebraic structure cubical sets are preferrable. Indeed, the composition of higher cells is not as nicely visible in simplicial sets as in cubical (or globular) sets. Also the definition of inverses to higher cells is much more algebraic in flavor, and algebraic tricks like subdivision into smaller cells works in a more obvious way.

Also, people who work with cubical sets have found it advantageous to introduce more degeneracy maps than just the obvious ones which "identify" opposite sides of the cube, e.g. such which identify neighbouring sides (e.g. as in ). This allows for more flexibility when folding cubes into the desired shapes, e.g. into balls as here.

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If I remember correctly, the proof of homotopy invariance of singular homology (at least the one in Hatcher) involves cutting a (simplex)x(interval) into simplices, which perhaps can be confusing. In cubical singular homology you'd just have to cut a (cube)x(interval) into cubes, which is obvious. This sort of thing maybe also comes up in other basic results that involve homotopies, such as (graded-)commutativity of cup product.

It doesn't really matter, since as you say, you get the same results at the end of the day. But perhaps in some sense simplices are more "basic" than cubes, since you can easily cut cubes (or any other polyhedron) into simplices, but you can't cut simplices into (finitely many) cubes or even "rectangle-ubes".

As for Ilya's answer, I actually don't think that cubes would necessarily complicate higher category theory stuff that much. It might even make certain things easier. Simplices are just a nice formalism with which to describe things like homotopies, and higher homotopies, and so forth. For example, if you have maps f, g, h such that (f compose g) and h are homotopic, you can think of the homotopy as being a triangle filling in the appropriate diagram. However, suppose you just have maps f and g that are homotopic. Then the homotopy is a "triangle" with a "degenerate" edge filling in the appropriate diagram. So simplices are not perfect either, we still need to allow for "degenerate" situations. In that case, we could have just started with cubes, allowing degenerate edges, to begin with...

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Thanks for your answer, Kevin. If I may be allowed a rather irrelevant comment, the fact that you can't cut a simplex into finitely many rectangle-ubes ( I like your terminology !) is surprisingly difficult. It is essentially Hilbert's third problem and was solved by Dehn only in 1900, with the help of his beautiful invariant. – Georges Elencwajg Nov 1 '09 at 13:29
Thanks for pointing that out! I remember now learning about Dehn's proof when I was an undergraduate. But I guess I had forgotten about it, as I just figured it was an "obvious" fact! – Kevin H. Lin Nov 1 '09 at 14:11
But aren't combinatorial (i.e. piecewise linear as opposed to globally linear) decompositions relevant here? The n-simplex can be divided into n+1 n-cubes. More specifically, if the n-simplex is the set of all (t_0, ..., t_n) with \sum t_i = 1, then the j-th cube is the subset where t_j > t_i for all i. – Kevin Walker Nov 1 '09 at 14:30
@Kevin W: Oh, yes, you are right. I was mixing up singular homology and simplicial homology in my head. So I guess that's one way you could prove equivalence of singular homology defined using simplicies and defined using cubes. – Kevin H. Lin Nov 1 '09 at 15:09
By the definition of simplices and cubes used for singular homology, each simplex is a cube (though rather degenerate one!). There's no need for Hibert's 3 problem at all :) I still think higher category theory is easier with simplices, but it's hard to explain why unless we both go seriously through the book and try to rewrite it using cubes, which probably isn't that important anyway... – Ilya Nikokoshev Nov 1 '09 at 18:33

I don't know for sure, but it would appear he means that it's easier to construct a cubic chain on a product X x Y given cubic chains on X and Y compared to the simplex chain given two simplex chains.

Since it's a simple exercise to go from simplices to cubes and vice versa, I don't see any advantage to cubes. In fact, I would expect any good book to explain that either of approaches could be used.

It would unnecessarily complicate the notation in the modern retellings of higher category theory, though. I learned about it from Lurie, Higher Topos Theory, 0608040 and there you really want to map simplices because you'll want to draw a picture of simplex [a_0, a_1, ..., a_n] being related to a composition of n categorical arrows, the arrow from a_0 to a_1 and so on.

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