Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in practical computation utilizing barycentric subdivisions is the size of $Bary(S)$ is much larger than $S$, and so when $S$ is large, then $Bary(S)$ is HUGE!
Has there been any research on "smaller" (i.e., having less simplices) simplicial models of $Bary(S)$ for general $S$ that maintain isomorphic homology groups, i.e., are homotopy equivalent?
I believe this would have much relevance to many areas of persistent homology and computational geometry.
Best Answer
Consider an arbitrary finite simplicial complex $\ S.\ $ First, look at it purely combinatorially. Thus, let $\ \{a\ b\}\ $ be a 1-simplex of $\ S.\ $ Then define a subdivided simplicial complex $\ S(a\ b\ c),\ $ where $\ c\ $ is a new fixed vertex that didn't belong to $\ S.\ $ The simplexes that don't contain $\ \{a\ b\}\ $ stay the same. Every simplex $$ \{a\ b\}\cup A $$ of $\ S\ $ that does contain $\{a\ b\} $ gets replaced by $\ \{a\ c\}\cup A\ $ and $\ \{c\ b\}\cup A.$
Now we can iterate this easy construction.
In turn, let's look at a geometric implementation $\ |S|,\ $ where we will have $\ c:=\frac{a+b}2.$
We may like to have iterations that have smaller and smaller mesh (mesh of a complex is the maximal length of 1-dimensional simplexes of that complex) so that mesh approaches zero.
We achieve this effect of mesh approaching zero by each time selecting 1-simplex $\ \{a\ b\}\ $ that has the maximal diamater among all present 1-simplexes (i.e. this diameter would be the current mesh). The mesh will approach zero slowly but it will.