classification theorem for compact surfaces is one of the great achievements of early 20th
century mathematics. The statement of this theorem is quite intuitive but it took about
sixty years until a rigorous proof was finally given by Brahana in 1921. Early
versions of the
classification theorem were given by Mobius in 1861, and by Jordan in 1866. More definite
proofs were given later by von Dyck in 1888 and Dehn and Heegaard in 1907.
This talk is
about the history of the theorem and the techniques used to prove it. We will give a
guided tour of the proof, pointing out which tools from algebraic topology are needed, and
give an abbreviated history of the
“proof.” A byproduct of the
theorem yields “global
parametrizations,” using fundamental domains,
a recent topic of research.