Completions and Ramifications

Abstract

We investigate ramifications, which are simplicial complexes constructed with a very simple inductive property: if two complexes are ramifications, then their union is a ramification whenever their intersection is a ramification. We show that the collection of all ramifications properly contains the collection of all collapsible complexes and that it is properly contained in the collection of all contractible complexes. We introduce the notion of a ramification pair, which is a couple of complexes satisfying also an inductive property. We establish a strong relation between ramification pairs and ramifications. In particular, the collection of ramification pairs is uniquely determined by the collection of ramifications. Also we provide some relationships between ramification pairs, collapsible pairs, and contractible pairs.

Appendix

In this appendix, we present a sequence of expansions and collapses which shows the contractibility of the dunce hat. We give this sequence for the reader who wants to better understand this object which is used several times in this paper for crucial counter-examples.

Let D be the triangulation of the dunce hat of Fig. 2(b). Let X be the cell whose facet is the set $\{3,5,6 \}$, thus $X \cap D$ is the closed curve that is highlighted in (c). Let $\gamma$ be the vertex corresponding to the label “1”, and let $E = \gamma X \cup D$.

The pair $( \{3,5,6\}, \{ \gamma , 3, 5 ,6 \})$ is a free pair for E, thus D is an elementary collapse of E. Let F be the complex given Fig. 2(c) and let $G = F \cup X$. It may be seen that E collapses onto G. First we remove the pair $( \{1,3,5\}, \{1,3,5,6\})$, then the pair $(\{ 1,5 \}, \{1,5,6 \})$, then the pair $( \{ 1,6 \}, \{1,6,3\}$. Now we observe that the complex G collapses onto the cell X, the first steps of a collapse sequence are represented Fig. 2(c). Since X is collapsible, the following sequence shows the contractibility of D:

$$D \nearrow E \searrow G \searrow X \searrow \emptyset $$