tailieunhanh - Relations among algebraic models of 1-connected homotopy 3-types

In this paper, we explore the relations among reduced cases of algebraic models for homotopy 3-types for groups such as braided crossed and quadratic modules and reduced simplicial groups with Moore complex of length 2. | Turk J Math 31 (2007) , 23 – 41. ¨ ITAK ˙ c TUB Relations Among Algebraic Models of 1-Connected Homotopy 3-Types Erdal Ulualan Abstract In this paper, we explore the relations among reduced cases of algebraic models for homotopy 3-types for groups such as braided crossed and quadratic modules and reduced simplicial groups with Moore complex of length 2. Key Words: Braided Crossed modules, Cat-groups, Simplicial groups, Quadratic Modules. 1. Introduction Whitehead [19] obtained an algebraic description of homotopy type of any 3-dimensional complex, and he gave the notion of crossed modules which model homotopy 2-type. Mac Lane used them to describe the third cohomology of a group, moreover, Mac Lane and Whitehead, [14], gave a description of 3-type in terms of a crossed module. Conduch´e [8] introduced the notion of 2-crossed module of groups model homotopy 3-type. Simplicial groups were studied by Kan [12]. Conduch´e also gave an equivalence between 2-crossed modules and simplicial groups with Moore complex of length 2. This equivalence establishes the role of 2-crossed modules as algebraic models of homotopy 3types since the homotopy properties of a simplicial group are given by its Moore complex. It is known that since crossed modules model homotopy 2-type, the category of crossed modules is equivalent to the category of simplicial groups with Moore complex of length 1. AMS Mathematics Subject Classification: 18D35, 18G30, 18G50, 18G55. 23 ULUALAN Brown and Gilbert [6] defined the braided, regular crossed modules which model homotopy 3-types. They proved that this structure is equivalent to the simplicial groups with Moore complex of length 2. This equivalence ensured that the braided, regular crossed modules model homotopy 3-types. Furthermore, they showed that the category of braided, regular crossed modules is equivalent to that of 2-crossed modules. The reduced case of braided, regular crossed module of groupoids is called a braided crossed module of

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