Irregularity in Complexes
Shaun Wylie · Annals of Mathematics · 1938
Introduction. In a complex Kn a simplex ois said to be regular if its star has the homology characters of an n-cell, the star being defined as the set of all simplexes of K having o-, as a face. If all the simplexes are regular K is said to be a manifold. This paper results from investigations of irregularity and of questions concerning the extent to which the irregularity of a simplex implies the irregularity of its faces. Answers are given to two main problems around which interest has crystallized: (i) Can K have all its simplexes regular except some of dimension p? (ii) Can K have all its simplexes regular except some of dimension > p? The second question arises because the answer to the first is in the negative unless p = 0 or n 1, and it can itself be answered in the affirmative; this can be seen by constructing an actual example for p = n 1, and by further subdividing it to introduce irregularity in lower dimensions. Lastly we make use of the technique of the earlier part to solve a problem left open in Tucker's thesis.' He finds three conditions on a cell-complex that it should be a manifold, closure, star, and intercept uniformity; but it is not there proved that intercept uniformity is independent of the other two. We construct an example of a complex having star and closure uniformity but lacking intercept uniformity, and thereby establish the required independence. The terminology of the paper is largely that of Lefschetz ;2 however, the term 'complement' is used instead of 'link complex'. I am indebted to Professor Tucker for the suggestion of the problems treated here, and also for some of the ideas involved in their solution.