Presentation of homotopy types under a space

TL;DR

This paper explores the relationship between homotopy types of spaces under a fixed space D and algebraic models, providing classifications and specific mappings in topological and algebraic categories.

Contribution

It introduces a comparison framework for homotopy types under a space D using algebraic models like crossed complexes and quadratic complexes, and classifies self-maps of S^2 x S^2.

Findings

Identifies subcategories of Top^D with algebraic categories

Shows there are exactly 16 essential self-maps of S^2 x S^2 fixing the diagonal

Establishes a correspondence between topological and algebraic models

Abstract

We compare the structure of a mapping cone in the category Top^D of spaces under a space D with differentials in algebraic models like crossed complexes and quadratic complexes. Several subcategories of Top^D are identified with algebraic categories. As an application we show that there are exactly 16 essential self--maps of S^2 x S^2 fixing the diagonal.