Completions and Ramifications
Résumé
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.
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