## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 4, Number 2 (2002), 191-218.

### 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 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.

#### Article information

**Source**

Homology Homotopy Appl. Volume 4, Number 2 (2002), 191-218.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

http://projecteuclid.org/euclid.hha/1139852462

**Mathematical Reviews number (MathSciNet)**

MR1918189

**Zentralblatt MATH identifier**

1065.55003

**Subjects**

Primary: 55P62: Rational homotopy theory

Secondary: 20F18: Nilpotent groups [See also 20D15]

#### Citation

Ekedahl, Torsten. On minimal models in integral homotopy theory. Homology Homotopy Appl. 4 (2002), no. 2, 191--218. http://projecteuclid.org/euclid.hha/1139852462.