information:extraction, structural:scholar_edition, linguistic:paraphrasing
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 topologically distinguished by examining their "holes." For instance, a figure-eight shape has more holes than a circle S 1 {\displaystyle S^{1}}, and a 2-torus T 2 {\displaystyle T^{2}} (a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere S 2 {\displaystyle S^{2}} (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 embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus T 2 {\displaystyle T^{2}} and 2-sphere S 2 {\displaystyle S^{2}} 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.