A homological interpretation of certain incidence matrices \(\text{mod }p\)
type: Item
instance of: scholarly article
title: A homological interpretation of certain incidence matrices \(\text{mod }p\)
zbMATH Open document ID: 0790.05016
DOI: 10.1007/BF01459502
author: T. Bier
published in: Mathematische Annalen
publication date: 1994-04-24T00:00:00Z
full work available at URL: https://eudml.org/doc/165132
review text: 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.
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 55U15
zbMATH DE Number: 537349
zbMATH Keywords: incidence matrices
zbMATH Keywords: boundary homomorphisms
zbMATH Keywords: simplex
zbMATH Keywords: homology groups
zbMATH Keywords: stable range
MaRDI profile type: Publication
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OpenAlex ID: W1977550062
description: scientific article; zbMATH DE number 537349