Topological Helly theorem

S. A. Bogatyi

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 365-405.

Résumé

We give an axiomatic version of topological Helly theorem, from which we derive many corollaries about common intersection (union). Instead of the space $\mathbb R^m$ we consider an arbitrary normal space $X$ with cohomological dimension not greater than $m$ and with trivial $m$-dimensional cohomological group. Instead of the convex subsets we consider closed acyclic subsets and instead of the conditions on intersections we impose (obtain) conditions on the values of arbitrary simple Boolean functions. In the extreme cases (only unions or intersections are considered) the conditions have the following form: for any $k$ sets of the given family, for $k\leq m+1$, either their common intersection has trivial cohomologies in all dimensions not greater than $m-k$, or their common union has trivial cohomologies in all dimensions from $\{k-2,\ldots,m-1\}$. Then it is proved that any subset obtained from sets of the given family with operations of union and intersection is nonempty and acyclic. For any closed covering of $m$-dimensional sphere the intersection of some $m+2$ elements is empty or for some $k\leq m+1$ there exist $k$ elements of the covering such that their intersection has non-trivial $(m+1-k)$-dimensional cohomologies. Our results are valid for arbitrary normal space of finite cohomological dimension, but are partially new even in the case of the plane. In particular, we fill the gap in the topological Helly theorem of 1930 for plane singular cells. If in the family of plane compacta the union of any 2 compacta is path-connected, and the union of any 3 compacta is simply connected, then the total intersection of all compacta of the family is non-empty. It is shown that if in the family of plane simply connected Peano continua the intersection of any 2 continua is connected and the intersection of any 3 continua is non-empty, then any compactum obtained from the compacta of the family with the operations of union and intersection is a non-empty simply connected Peano continuum. Analogously, if in the family of plane simply connected Peano continua the union of any 2 and any 3 continua is a simply connected Peano continuum, then any compactum obtained from the compacta of the family with the operations of union and intersection is a non-empty simply connected Peano continuum. Analogous statements are true for continua that do not separate the plane.

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