A homological interpretation of certain incidence matrices \(\text{mod }p\)
We define the \(k\)-step boundary homomorphisms of the simplex by the 0-1- incidence matrices of the \(q\)-subsets versus the \((q-k)\)-subsets of an \(n\)-set. For a given prime \(p\) and suitable step lengths \(k'\), \(k''\) we obtain homology groups of the simplex. These groups can be calculated in many cases. In this paper we regard the range in which the size of the chains \(q\) is ``not too close to the centre \({n\over 2}\)'', a condition to be called the stable range, in which case we can completely calculate these homology groups. As a special case we obtain the result that mod 2 the \(n\)-simplex is acyclic in the stable range iff the step length is a power of 2. Another special case for an odd prime \(p\) under the condition \(k'+k''=p\) had been considered more than 40 years ago by Mayer and Spanier.
scientific article; zbMATH DE number 537349