Remarks on pointed digital homotopy

We present and explore in detail a pair of digital images with $c_u$-adjacencies that are homotopic but not pointed homotopic. For two digital loops $f,g: [0,m]_Z \rightarrow X$ with the same basepoint, we introduce the notion of {\em tight at the basepoint (TAB)} pointed homotopy, which is more restrictive than ordinary pointed homotopy and yields some different results. We present a variant form of the digital fundamental group. Based on what we call {\em eventually constant} loops, this version of the fundamental group is equivalent to that of Boxer (1999), but offers the advantage that eventually constant maps are often easier to work with than the trivial extensions that are key to the development of the fundamental group in Boxer (1999) and many subsequent papers. We show that homotopy equivalent digital images have isomorphic fundamental groups, even when the homotopy equivalence does not preserve the basepoint. This assertion appeared in Boxer (2005), but there was an error in the proof; here, we correct the error.