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Why can some loops on a torus not be shrunk to a point?

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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, the ability to shrink a loop to a point is a defining characteristic of a space being "simply connected." While a sphere is simply connected because any loop on its surface can be continuously deformed and shrunk to an arbitrarily small diameter, a torus—the surface of a doughnut—is not.1

The reason some loops on a torus cannot be shrunk to a point is due to the object's fundamental geometric structure, which contains "holes" that prevent certain loops from being contracted without leaving the surface or tearing it. Specifically, a loop that wraps around the central hole of the torus or around the tube itself is obstructed by the shape of the surface itself. Because homeomorphism preserves these topological properties, this inability to shrink such loops distinguishes the torus from simply connected shapes like the sphere.12