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How can shrinking loops reveal whether a surface is simply connected?

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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 topology, a surface is considered simply connected if every closed loop drawn on it can be continuously shrunk to a single point without leaving the surface.1

This process relies on the concept of homotopy, which classifies geometric regions based on the behavior of paths within them. If a surface has no "holes," any loop can be deformed (shrunk) into an arbitrarily small diameter, effectively becoming a point. In such cases, all closed paths are considered homotopic to one another, and the fundamental group of the space consists of a single, trivial element.213

Conversely, if a surface contains a hole—such as the surface of a torus—certain loops cannot be shrunk to a point because they are obstructed by the hole. These loops cannot be continuously deformed into a point without passing through the hole and leaving the defined region, indicating that the surface is not simply connected.12

Would you like to learn more about how the fundamental group classifies these different types of surfaces?