Homology, Homotopy and Applications

Volume 7 (2005)

Number 2

Proceedings of a Special Session of a Joint RSME-AMS Meeting at Sevilla University

Algebraic models for homotopy types

Pages: 139 – 160

DOI: http://dx.doi.org/10.4310/HHA.2005.v7.n2.a8

Authors

Julio Rubio (Departamento de Matemáticas y Computación, Universidad de La Rioja, Spain)

Francis Sergeraert (Institut Fourier, Université Joseph Fourier, St Martin d’Hères, France)

Abstract

The classical problem of algebraic models for homotopy types is precisely stated here in terms of our ability to compute with the models. Two different natural statements for this problem are produced, the simplest one being entirely solved by the notion of $\mathcal{SS_{EH}}$-structure, due to the authors. Other tentative solutions, Postnikov towers and $E_{\infty}$-chain complexes, are considered and compared with the $\mathcal{SS_{EH}}$-structures. In particular, an imprecision in the usual definition of the $k$-“invariants” is explained, which implies we seem far from a solution for the ideal statement of our problem. On the positive side, the problem stated below in the framed quotation is solved.

Keywords

classification, homotopy type, Postnikov system, k-invariant, computable model, algebraic model, computability

2010 Mathematics Subject Classification

Primary 55P15. Secondary 18G40, 18G55, 55S45.

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