  Generic polyhedra are interesting mathematical objects to study in their own
right. In this paper, we initialize a systematic study of two-dimensional
generic polyhedra with an eye towards applications to low-dimensional topology,
especially the Andrews-Curtis and Zeeman conjectures. After recalling the basic
notions of generic polyhedra and fake surfaces, we derive some interesting
properties of fake surfaces. Our main result is a complete classification of
acyclic cellular fake surfaces up to complexity 4 and a classification of
acyclic cellular fake surfaces without small disks of complexity 5. From this
classification, we prove the contractibility conjecture for acyclic cellular
fake surfaces of complexity 4, and the embedded disk conjecture up to
complexity 5. We provide evidence for the conjectures that the probability of
being a spine among fake surfaces is 0 and that every contractible fake surface
has an embedded disk.
