I am reading Bridson and Haefliger's ``Metric Spaces of Non-Positive Curvature" and I am struggling with Definition 5.9 of a free face.
Let $K$ be an $M_\kappa$-polyhedral complex. A closed $n$-cell $B$ in $K$ is said to be a free face if it is contained in the boundary of exactly one cell $B'$ of higher dimension and the intersection of $B'$ with some small neighborhood of an interior point of $B$ is connected.
My first question is, from this definition there does not appear to be a restriction on codimension, however, in the `usual' combinatorial definition, codimension 1 is required.
Secondly, in the following complex
are all three edges ($a$, $b$ and $c$) free faces?
Finally, can you give an examples of non trivial contractible and non contractible simplicial complexes with no free faces? (For the latter I think a regular hollow tetrahedron works.)
Yes, all the edges in your example are free faces. In contrast, vertices are not free faces. I am not sure where did you find the "usual combinatorial definition", but if you consider the disjoint union of your example with a simplex of dimension 3 then edges are no longer of codimension 1 but they are still free.
Dunce hat is a classical example of a contractible complex without free faces. It is only a Delta-complex not a simplicial complex. If this bothers you, just apply the barycentric subdivision twice. Another standard example is Bing's house with two rooms.