Editing Homology (mathematics) (section)

=== Inspirations for homology (informal discussion) ===
One of the ideas that led to the development of homology was the observation that certain low-dimensional shapes can be [[Topology|topologically]] distinguished by examining their "holes." For instance, a figure-eight shape has more holes than a circle <math>S^1</math>, and a 2-torus <math>T^2</math> (a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere <math>S^2</math> (a 2-dimensional surface shaped like a basketball).

Studying topological features such as these led to the notion of the ''cycles'' that represent homology classes (the elements of homology groups). For example, the two [[Embedding|embedded]] circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus <math>T^2</math> and 2-sphere <math>S^2</math> represent 2-cycles. Cycles form a group under the operation of ''formal addition,'' which refers to adding cycles symbolically rather than combining them geometrically. Any formal sum of cycles is again called a cycle.