A COMBINATORIAL DECOMPOSITION OF ACYCLIC SIMPLICIAL COMPLEXES

It is proved that if DELTA is a finite acyclic simplicial complex, the n there is a subcomplex DELTA' subset-of DELTA and a bijection eta: DE LTA' --> DELTA - DELTA' such that F subset-of eta(F) and \eta(F)-F\ = 1 for all F is-an-element-of DELTA'. This improves an earlier result o f Kalai. An immediate corollary is a characterization (first due to Ka lai) of the f-vector of an acyclic simplicial complex. Several general izations, some proved and some conjectured, are discussed.