I am familiar with the usual definition of simplicial complex or abstract simplicial complex. However, while reading Lyndon and Schupp's Combinatorial Group Theory, p. 117, the following statement is made:

If $G = (X;R)$ is the multiplication table presentation in which every element of R has length 3, then every face of $K = K(X;R)$ is a triangle and K is simplicial.

Where K is the usual complex associated to a group presentation.

My question: How is K simplicial when it only has a single vertex? The usual definitions for simplicial complexes define it in terms of subsets of the powerset of the vertex set, subject to some constraints. But this doesn't work if the vertex set is a singleton! Is there a standard rigorous alternative definition used in combinatorial group theory?

Edit for Clarification (Lyndon and Schupp's definition of the complex K):

With every presentation $G = (X,R)$ where all r in R are cyclically reduced we associate a special complex $K(X;R)$ with a single vertex. First, K has a single vertex v, and an edge $\tilde{x}$ (from v to v) for each element x of X, together with its inverse $\tilde{x}^{-1}$. Now every path in K is a loop. Second, if $r = x_1^{e_1} \cdot \cdot \cdot x_n^{e_n}$ is in R where all $x_i \in X$, $e_i = \pm 1$, we introduce a face $D$ with boundary path (at v) $\tilde{x}_1^{e_1} \cdot \cdot \cdot \tilde{x}_n^{e_n}$, together with $D^{-1}$.

  • $\begingroup$ How do you get only a single vertex for $K$? Aren't the vertices all of the elements of $G$? $\endgroup$ – Steven Stadnicki Mar 15 '17 at 17:50
  • $\begingroup$ It is possible that the book means "simplicial set" rather than "abstract simplicial complex." But, again, it is hard to guess. $\endgroup$ – Thomas Andrews Mar 15 '17 at 18:00
  • $\begingroup$ Given your update, I definitely think the book means "simplicial set." $\endgroup$ – Thomas Andrews Mar 15 '17 at 18:13

I think that what Lyndon and Schupp are doing can be easily formalized using the concept of a "Delta complex" as Hatcher calls it in his Algebraic Topology textbook.

A Delta complex $X$ is a CW complex that shares some features with a simplicial complex, in that each $n$-simplex is the image of a characteristic map $\sigma : \Delta^n \to X$ where $\Delta^n$ is the "standard" $n$-simplex. Furthermore, the restriction of $\sigma$ to each open $k$-dimensional face of $\Delta^n$ is required to be a homeomorphism onto the interior of some $k$-dimensional simplex of $X$. However, the characteristic maps $\sigma$ are not required to be embeddings, nor even local embeddings. So, for example, it is perfectly fine in a Delta complex for there to be only one $0$-dimensional cell, so $\sigma : \Delta^2 \to X$ will never be an embedding. In fact it's also fine to have length 3 relators of the form $a a^{-1} b$ which leads to a characteristic map $\sigma : D^2 \to X$ that is not a local embedding.

This does beg the question of what Lyndon and Schupp meant when they wrote that $K$ is simplicial, because their book is decades older than Hatcher's, and I'm pretty sure that the "Delta complex" terminology was first used by Hatcher.

Although I cannot answer that question without being able to look up what Lyndon and Schupp wrote, I'll say that before there were Delta complexes, topologists loosely used the terminology "cell complex" to describe a CW complex in which the characteristic maps $\sigma : D^n \to X$ had a property similar to the one for a Delta complex: the boundary sphere $S^{n-1}$ was subdivided into an $n-1$ dimensional cell complex such that $\sigma | S^{n-1}$ maps each open cell of $S^{n-1}$ homeomorphically to an open cell of $X$ (thus making this a definition-by-induction-on-dimension).

In the setting of Lyndon and Schupp, the complex is $2$-dimensional and the boundary subdivision of each characteristic map $\sigma : D^2 \to X$ has exactly three 1-cells, making $D^2$ cellularly isomorphic to $\Delta^2$. This is probably pretty close to what they were thinking in referring to $X$ as "simplicial".


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