Computational homotopy of finite regular CW-spaces

We describe a computational approach to the basic homotopy theory of finite regular CW-spaces, paying particular attention to spaces arising as a union of closures of n-cells with isomorphic face posets. Deformation retraction methods for computing fundamental groups, integral homology and persistent homology are given in this context. The approach is not new but our account contains some new features: (i) certain computational advantages of a permutahedral face poset are identified and utilized; (ii) the notion of lattice complex is introduced as a data type for implementing a general class of regular CW-spaces (including certain pure cubical and pure permutahedral subspaces of flat manifolds such as ℝⁿ); (iii) zig-zag homotopy retractions are introduced as an initial procedure for reducing the number of cells of low-dimensional lattice spaces; more standard discrete vector field techniques are applied for further cellular reduction; (iv) a persistent homology approach to feature recognition in low-dimensional digital images is illustrated; (v) fundamental groups are computed; (vi) algorithms are implemented in the gap system for computational algebra, allowing for their output to benefit from the system's vast library of efficient algebraic procedures.