Abstract: Joint project with William Jaco. Following the steps for the prime
decomposition of 3-manifolds laid out in the Jaco & Rubinstein paper ``$0$-
Efficient Triangulations of 3-Manifolds'' we construct the analogous (and
well known) result for surfaces. That being the torus and real projective
plane are the prime surfaces with respect to the connect sum operation
on surfaces. Further the decomposition of surfaces into prime elements
can be written uniquely up to homeomorphism.
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