On Combinatorial Topology
Albert William Tucker · Proceedings of the National Academy of Sciences · 1932
This note gives a preliminary account of some features of a paper on combinatorial topology to be published in more complete detail at a later date.The terminology and notation are patterned after that of Lefschetz,1 but the cells {EpI with which we shall deal are of the ab- stract type considered by Mayer.2The advantage of using a general type of cell where the incidence numbers 77 may have any integral values, is described by Lefschetz (loc.cit., pp.104-5).But there is one difficulty, viz., an Ep cancelling out of the chain-boundary of an Ep + i is not (combi- natorially) distinguished from an Ep not on the boundary of Ep + 1 at all.This can be overcome by introducing a relation telling when an Ep is on the boundary of an E,(p < q).The only connection of this quali- tative relation with the quantitative one of incidence numbers is that if Ep is not on the boundary of Ep + 1 the incidence number is 0. 1. Open and Closed Sub-Complexes.-A complex K is a set of cells such that the boundary-chain of a boundary-chain is always 0, i.e.,