Computational homology of n-types

Abstract

An n-type X is a CW-space with homotopy groups \pi_iX=0 for all i>n. Up to homotopy equivalence such a space can be specified algebraically by means of a simplicial group whose Moore complex is trivial in degrees >= n. For the case n=2, Mac Lane and Whitehead [1950] showed that there is a one-one correspondence between 2-types and quasi-isomorphism classes of crossed modules. The main goal of this PhD thesis is the development of computational tools for helping with the classification of 2-types. Our primary computational tool is the homology, and persistent homology, of 2-types. We provide a classification of most of the 2-types of order m <= 255.