Polynomial-Time Homology for Simplicial Eilenberg–MacLane Spaces

Abstract

In an earlier paper of Čadek, Vokřínek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the spaces. We aim at showing that, if the dimensions of the considered spaces are bounded by a constant, then the computations can be done in polynomial time. In this paper we make a significant technical step towards this goal: we show that the Eilenberg–MacLane space $K(\mathbb{Z},1)$ , represented as a simplicial group, can be equipped with polynomial-time homology (this is a polynomial-time version of effective homology considered in previous works of the third author and co-workers). To this end, we construct a suitable discrete vector field, in the sense of Forman’s discrete Morse theory, on $K(\mathbb{Z},1)$ . The construction is purely combinatorial and it can be understood as a certain procedure for reducing finite sequences of integers, without any reference to topology. The Eilenberg–MacLane spaces are the basic building blocks in a Postnikov system, which is a “layered” representation of a topological space suitable for homotopy-theoretic computations. Employing the result of this paper together with other results on polynomial-time homology, in another paper we obtain, for every fixed k, a polynomial-time algorithm for computing the kth homotopy group π k (X) of a given simply connected space X, as well as the first k stages of a Postnikov system for X, and also a polynomial-time version of the algorithm of Čadek et al. mentioned above.