  The current paper focuses on fundamental groups and Euler characteristics of
various digital models of the 2-dimensional sphere. For all models that we
consider, we show that the fundamental groups are trivial, and compute the
Euler characteristics (which are not always equal). We consider the connected
sum of digital surfaces and investigate how this operation relates to the
fundamental group and Euler characteristic. We also consider two related but
different notions of a digital image having "no holes," and relate this to the
triviality of the fundamental group.
  Many of our results have origins in a paper by S.-E. Han, which contains many
errors. We correct these errors when possible, and leave some open questions.
We also present some original results.
