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# Homology, Homotopy and Applications

## Volume 4 (2002)

### Number 2

### The Roos Festschrift volume

### On minimal models in integral homotopy theory

Pages: 191 – 218

DOI: http://dx.doi.org/10.4310/HHA.2002.v4.n2.a9

#### Author

#### Abstract

This paper takes its starting point in an idea of Grothendieck on the representation of homotopy types. We show that any locally finite nilpotent homotopy type can be represented by a simplicial set which is a finitely generated free group in all degrees and whose maps are given by polynomials with rational coefficients. Such a simplicial set is in some sense a universal localisation/completion as all localisations and completions of the homotopy is easily constructed from it. In particular relations with the Quillen and Sullivan approaches are presented. When the theory is applied to the Eilenberg-MacLane space of a torsion free finitely generated nilpotent group a close relation to the the theory of Passi polynomial maps is obtained.

#### Keywords

homotopy, integral models, passi polynomial maps

#### 2010 Mathematics Subject Classification

14Axx, 20F18, 55P15, 55P60, 55P62, 55U10