Some Remarks on the Join of Two Complexes and on Invariant Subsets of a Complex
Egbert R. Van Kampen · American Journal of Mathematics · 1932
In this paper we consider some miscellaneous properties of combinatorial topology mostly found during the preparation of a course on this subject. The first three sections deal with the homology characters of a join; it is found useful to introduce homology characters in a combinatorial way for a kind of open complex slightly more general than is customary. After that we prove some simple properties of invariant subsets of complexes, soon concentrating on the problem of dividing any complex into the smallest invariant subsets that can be found with the methods rnow at the disposal of combinatorial topology. To do this we! have to prove a number of properties on the invariance in a complex of the invariants of the neighborhood complexes of the different simplexes of the complex, using some of the properties of joins proved previously. 1. As the cells of the join (K1 K2) and the product (K1 X K2) of two complexes (K1) and (K2) as well as their incidence relations are completely determined by K1 and K2 it is a matter of calculation to find their homology characters when K1 and K2 are given. Superficially the computation seems easier for the product because in the product there is one (p, + q) cell corresponding to every pair formed by a pLcell of K1 and a q-cell of K2 and no other cell; the boundary of the (p + q) -cell corresponds to all the pairs which can be formed by replacing the p,or the q-cell by its boundary cells. The structure of a join is slightly more complicated. This induced A. B. Browin * to reduce the computation for K1 K2 to that for K1 X K2 by means of a relatively complicated reasoning on different kinds of chains on the join; he does not find any result on torsion numbers. However, when we introduce formally a (1) -dimensional simplex into all the relevant complexes forming the boundary of all vertices and such that its join with any cell is that cell, then there is in the join a (p+ q + 1)-simplex corresponding to every pair formed by a p-simplex of K1 and a q-simplex of K2; with analogous boundary relations as for the product. The result for the product t can now be transcribed for the join without any additional calculation.