Question: Why do we use continuous maps from the standard $n$-simplices $\Delta_n$, and not another space like $B^n$ or $I^n$, when defining singular homology?
I am taking courses in convex optimization and homology theory at the same time. I was expecting what I learned in convex analysis about simplices to be relevant and applied frequently to the study of singular homology, since the latter is defined in terms of simplices, but this does not seem to be the case. Thus I am confused about why simplices are used in defining singular homology in the first place, when singular homology does not seem to take advantage of the properties of simplices.
For example, higher-dimensional cubes $I^n$ or closed balls $B^n$ (I'm not sure which) are used when defining the higher homotopy groups, and for $n=0,1$ the standard simplices, cube, and ball all coincide (point and unit interval, respectively). This allows one to interpret $0$- and $1$-chains in a space $X$ as points and/or paths in $X$, respectively. However, $\Delta_n$ and $I_n$ (or $B_n$) no longer coincide for $n \ge 2$, so it is unclear whether the analogous identifications still hold, see e.g. here.
Likewise, one uses closed spheres $B^n$ for defining CW complexes, so if there was a theory using continuous maps from the $B^n$ (instead of from the $\Delta^n$), perhaps it would be easier to apply to the important class of spaces that CW complexes?
In other words, it seems like there is a big tradeoff for using the simplices $\Delta_n$ instead of the more natural choices of spheres $B^n$ or cubes $I^n$, so there should be a good reason justifying this tradeoff — however none of my books address this issue, leaving me very confused.
Also, simplices play a very big role in convex analysis (to the best of my knowledge), and can be shown within the framework of convex analysis to have many special properties. However, when working with singular homology, it never seems like anyone ever uses these special properties as identified in convex analysis — indeed, I have never seen an algebraic topology book with an interlude or appendix explaining facts from convex analysis relevant to understanding simplices (and thus seemingly also singular homology) — why is that?
Likewise, in convex analysis one also studies cubes and cross-polytopes as well as their images under affine transformations. Why don't we also consider continuous maps from those spaces?
One can also define in addition to simplicial complexes complexes from cubes (cubical complexes) as well as "cell complexes built from arbitrary compact convex polytopes [which] are likewise easy to work with" (Ghrist, Elementary Applied Topology, p. 27). Why don't we ever work with continuous maps from arbitrary compact convex polytopes (including cubes and cross-polytopes) or continuous maps from arbitrary convex bodies (compact and convex spaces, including spheres)?
From Massey, A Basic Course in Algebraic Topology, p. 148:
If $X$ is a compact, connected, orientable n-dimensional manifold, then $H_n(X)$ is infinite cyclic, and $H_q(X)=\{0\}$ for all $q>n$. In some vague sense, such a manifold is a prototype or model for nonzero $n-$dimensional homology groups.
Would considering continuous maps from $I^n$ or $B^n$ or any other graded sequence of compact convex spaces not produce a homology theory, the way continuous maps from simplices $\Delta_n$ do? Even if they did, would these homology theories not be homotopy invariant?
Best Answer
nLab has a page on cubical sets with connections with the following excerpt:
What I take away from this is that, while cubical sets do correctly capture the homotopy type of a space, their category is defective in a way that the category of simplicial sets is not, and the foundations of the subject were developed before it was discovered how to repair the notion of cubical set.