Strong homotopy of digitally continuous functions

Abstract

We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We explore basic properties of strong homotopy, and give some equivalent characterizations. In particular we show that strong homotopy is related to ``punctuated homotopy,'' in which the function changes by only one point in each homotopy time step. We also show that strongly homotopic maps always have the same induced homomorphisms in the digital homology theory. This is not generally true for digitally homotopic maps, though we do show that it is true for any homotopic selfmaps on the digital cycle $C_n$ with $n\ge 4$. We also define and consider strong homotopy equivalence of digital images. Using some computer assistance, we produce a catalog of all small digital images up to strong homotopy equivalence. We also briefly consider pointed strong homotopy equivalence, and give an example of a pointed contractible image which is not pointed strongly contractible.