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How did Max Dehn and Poul Heegaard contribute to the classification of surfaces?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In the context of topology, the German-born mathematician Max Dehn and the Danish mathematician Poul Heegaard made a foundational contribution to the field by jointly presenting one of the first classification theorems for two-dimensional surfaces in 1907.1

Their work was a significant development within combinatorial topology, a branch of the field that focuses on understanding topological properties by combining elementary shapes like polygons or polyhedra. By building upon the earlier foundations of Henri Poincaré, who had established a context for applying algebraic ideas to topological objects, Dehn and Heegaard helped formalize the systematic categorization of surfaces. This research was instrumental in the early 20th-century effort to classify topological manifolds, a pursuit that also led Dehn to develop his own namesake theorem, Dehn’s lemma, in 1910.12