Let $X$ be a simplicial complex on a vertex set $V=\{1,\dots,n\}$. Suppose there is some subset $U\subseteq V$ such that every facet of $X$ contains $U$ or is disjoint with $U$.

Then, define $X'$ on a new vertex set $V'=(V-U)\cup \{v_U\}$ and facets $\sigma'$ where $\sigma'=\sigma$ if $\sigma$ is a facet of $X$ disjoint to $U$, and $\sigma'=(\sigma-U)\cup \{v_U\}$ if $\sigma$ is a facet of $X$ containing $U$. Then $X$ and $X'$ have the same homology.

I was able to prove this by cases (There are a few details I haven't written at the end so it might have some mistakes, but I don't think so): First, if every facet contains $U$, then I used a Mayer-Vietoris sequence to prove there is no homology in this case. Second, if no facet contains $U$ (Thus every facet is disjoint to $U$), then the simplicial complex remains unchanged by the operation, therefore there is also no change in its homology.

Lastly, in the general case, I used Mayer-Vietoris sequences for both $X$ and $X'$ taking $X=A\cup B,X'=A'\cup B$ where $B$ consists in the facets disjoint to $U$ (Or not containing $v_U$ in $X'$) and $A,A'$ consist in the facets containing $U,v_U$ respectively. After using the fact that the homologies of $A,A'$ are zero, and that the intersections $A\cap B=A'\cap B$ remain unchanged by the operation it's easy to define a chain isomorphism between the exact sequences by the five lemma (Where four of the morphisms are identity maps and the fifth one is induced by the inclusion $v_U\mapsto v$ for any fixed $v\in U$).

My question is about the following remarks:

First, the operation done is analogous to the operation of edge contraction. We can interpret it in some ways. It identifies the vertices in $U$, and faces not disjoint with $U$, causing some of them to reduce their dimension (If they contain more than one vertex of $U$). In this case if I'm not mistaken we can also interpret it like removing all the vertices of $U$ but one.

Second: It doesn't always preserve homology, but sometimes it does. In this case the map $v_U\mapsto v$ (for a fixed $v\in U$) even seems to be a good candidate for a homotopy equivalence.

My question is the following: Is there any literature about the "edge contraction" operator applied to simplicial complexes? It seems like a rigid version of Proposition 0.17 in Hatcher's book.


1 Answer 1


$\newcommand{\cU}{\mathcal{U}}$ This actually is related to the construction of the "Nerve" of a simplicial complex.

Let $\Delta$ be a simplicial complex with facet set $F(\Delta)=\{\sigma_1,\dots,\sigma_r\}$ and vertices $v_1,\dots,v_n$. For each facet of $\Delta$, consider a small enough open set around it. Now you have an open cover $\cU=\{U_1,\dots,U_r\}$. The nerve $N=N(\cU)$ is defined to be a simplicial complex with vertex set $\cU$ given by $N=\{\tau\subseteq\cU:\cap\tau\neq\varnothing\}$, i.e. the subcollections of open sets with nonempty intersection.

You can make a similar discrete construction, by setting a complex $D=D(\Delta)$ with vertices $V=\{\sigma_1,\dots,\sigma_r\}$ defined by $D=\{\tau\subseteq V: \cap \tau\neq \varnothing\}$.

You can choose $\cU$ appropiately, so that the open sets are not "too big" so that they only intersect each other if the corresponding facets do: enter image description here And in this case, it's clear that $N$ and $D$ are isomorphic by the map $\sigma_i\mapsto U_i$. So we'll call $D$ the nerve of $\Delta$.

Now, let $D^2$ be $D(D(\Delta))$. Naively, the facets of $D$ are given by $F(D)=\{\tau_1,\dots,\tau_n\}$ where $\tau_i=\{\sigma\in F(\Delta):v_i\in\sigma\}$, so the vertices of $D^2$ are $\tau_1,\dots,\tau_n$. The facets of $D^2$ are thus $\rho_1,\dots,\rho_r$ where $\rho_j=\{\tau\in F(D): \sigma_j\in\tau\}$.

Now, $\sigma_j\in \tau_i$ iff $v_i\in \sigma_j$, therefore after renaming $\tau_i$ by $v_i$ you get $D^2=\Delta$.

Is this correct? Actually, it's not. We're missing that $\tau_i$ may equal $\tau_{i'}$ for $i\neq i'$, so $D^2$ will be a little smaller than $\Delta$. When you remove repeated elements, you'll find that $D^2$ is what results of $\Delta$ after contracting all the vertex sets with the condition that every facet of $\Delta$ contains it or is disjoint with it. And then you have the Nerve Theorem of Leray which proves that $D$ is homotopically equivalent to $\Delta$ which means that of course $D^2$ is also homotopically equivalent to it. I expect the proof of that theorem should help to prove that the intermediate steps of contracting those sets one by one also preserve homotopy.


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