Homotopy 2-Types of Low Order

• Published in: Experimental mathematics 2014-10, Vol.23 (4), p.383-389
• Main Authors: Ellis, Graham, Le, Luyen Van
• Format: Article
• Language: English
• Subjects: crossed module; homotopy 2-type; quasi-isomorphism
• ISSN: 1058-6458; 1944-950X
• DOI: 10.1080/10586458.2014.912059

There is a well-known equivalence between the homotopy types of connected CW-spaces X with π n X = 0 for n ≠ 1, 2 and the quasi-isomorphism classes of crossed modules ∂: M → P [ Mac Lane and Whitehead 50 ]. When the homotopy groups π 1 X and π 2 X are finite, one can represent the homotopy type of X by a crossed module in which M and P are finite groups. We define the order of such a crossed module to be |∂| = |M| × |P|, and the order of a quasi-isomorphism class of crossed modules to be the least order among all crossed modules in the class. We then define the order of a homotopy 2-type X to be the order of the corresponding quasi-isomorphism class of crossed modules. In this paper, we describe a computer implementation that inputs a finite crossed module of reasonably small order and returns a quasi-isomorphic crossed module of least order. Underlying the function is a catalog of all quasi-isomorphism classes of order m ≤ 127, m ≠ 32, 64, 81, 96, and a catalog of all isomorphism classes of crossed modules of order m ≤ 255.