Homotopy 2-Types of Low Order

Abstract

There is a well-known equivalence between the homotopy types of connected CW-spaces X with π_nX = 0 for n ≠ 1, 2 and the quasi-isomorphism classes of crossed modules ∂: M → P [Mac Lane and Whitehead 50]. When the homotopy groups π₁X and π₂X 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.

Keywords::

• crossed module
• homotopy 2-type
• quasi-isomorphism
• 55P15