I have been reading about the topological invariance of simplicial homology groups, and at one point a weaker result is needed, namely the invariance of homology groups under barycentric subdivision. I've been reading both Armstrong's "Basic Topology" and Giblins's "Graphs, Surfaces and Homology" and both take the same approach to prove this fact: they obtain the barycentric subdivision of a complex by repeated application of a simpler operation called stellar subdivision. The required definitions are as follows:

  • The barycentre of an $n$-simplex $\sigma=(v_0,...,v_n)$ is the point $\widehat{\sigma}=\frac{1}{n+1}(v_0+\dots+v_n)\in\sigma$.

  • Given a simplicial complex $K$, the cone of a subcomplex $L$ with vertex $v\notin |L|$ is the complex comprised of $v$, the simplexes of $L$, and the simplexes created by adding to each of the latter the vertex $v$.

  • The first barycentric subdivision of $K$, denoted $\boldsymbol{K^1}$, is the simplicial complex created by replacing, in order of increasing dimension, each simplex $\sigma\in K$ by the cone of its boundary with vertex its barycentre $\hat{\sigma}$.

  • The simplical complex $K'$ obtained from $K$ by stellar subdivision of $\sigma$ is the one in which the simplexes of $K$ which do not have $\sigma$ as a face are left untouched, while for each $\sigma'\in K$ with $\sigma$ as a face, we consider $L$ the subcomplex of the boundary of $\sigma'$ comprised of the simplexes which do not have $\sigma$ as a face, and replace $\sigma'$ by the cone of $L$ with vertex $\hat{\sigma}$.

Then it is stated that the barycentric subdivision of $K$ can be obtained by stellar-subdividing each of its simplexes in order of decreasing dimension. Both books give a visual example to convince the reader this statement is true, but neither of them gives a formal proof of the fact.

At first, I did not know why all of this was even necessary, since we already had a method of obtaining the barycentric subdivision of $K$ by repeated application of a simple operation as explained in the definition, but then I realized that after applying the operation described to a single simplex, the result is not necessarily a simplicial complex.

So now, I'd like to formally prove that the method described using stellar subdivisions does indeed yield the desired barycentric subdivision. I'm at a loss at how to do that, though, since I don't know how I can prove two simplicial complexes obtained by so different methods are the same.

Thanks for reading!


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