Hole

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a disconnectivity is interpreted as a hole in the space. Examples of holes are things like the "donut hole" in the center of the torus, a domain removed from a plane, and the portion missing from Euclidean space after cutting a knot out from it.

Singular homology groups form a measure of the hole structure of a space, but they are one particular measure and they don't always detect all holes. homotopy groups of a space are another measure of holes in a space, as well as bordism groups, K-theory, cohomotopy groups, and so on.

There are many ways to measure holes in a space. Some holes are picked up by homotopy groups that are not detected by homology groups, and some holes are detected by homology groups that are not picked up by homotopy groups. (For example, in the torus, homotopy groups "miss" the two-dimensional hole that is given by the torus itself, but the second homology group picks that hole up.) In addition, homology groups don't detect the varying hole structures of the complement of knots in three-space, but the first homotopy group (the fundamental group) does.