On minimal models in integral homotopy theory

Abstract

This paper takes its starting point in an idea of Grothendieck on the representation of homotopy types. We show that any locally Þnite nilpotent homotopy type can be represented by a simplicial set which is a Þnitely generated free group in all degrees and whose maps are given by poly- nomials with rational coecients. Such a simplicial set is in some sense a universal localisation/completion as all localisa- tions and completions of the homotopy is easily constructed from it. In particular relations with the Quillen and Sulli- van approaches are presented. When the theory is applied to the Eilenberg-MacLane space of a torsion free Þnitely gener- ated nilpotent group a close relation to the the theory of Passi polynomial maps is obtained. To Jan-Erik Roos on his sixty-Þfth birthday Inspired by some ideas of A. Grothendieck ((5)) I shall in this article give an algebraic description of nilpotent homotopy types with Þnitely generated homology (in each degree). From a homotopy theoretic perspective the main result says that every such nilpo- tent homotopy type may be represented by a simplicial set which is of the form Z n for some n in each degree and for which the face and degeneracy maps are numerical maps; maps that are given by polynomials with rational coecients. Furthermore, the cohomology may be computed using numerical cochains, any map between such models is homotopic to a numerical one and homotopic numerical maps are numer- ically homotopic. The construction of a model is rather straightforward; one Þrst shows that the cohomology of an Eilenberg Mac-Lane space can be computed using numerical cochains and then uses induction over a principal Postnikov tower. Localisation and completion Þts very nicely into this framework. If R is either a subring of Q or is the ring of p-adic numbers for some prime p and K := R N Q then a numerical function Z m ! Z n clearly induces a map K m ! K n but also takes R m to R n . Hence a numerical simplicial set gives rise to a numerical set obtained by replacing each Z n that appears in some degree by R n . For a model this new space is the R-localisation when R is a subring of Q and the p-completion when R = Zp. In this way a numerical model might be thought of as a universal localisation.