on the distinguished point, and composition with the pinch gives the sum map. More generally, the -th homotopy group, begins with the pointed -sphere , and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for equal to (for positive ) — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some all maps are null homotopic, then the group consists of one element, and is called the trivial group. A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism), so that the two spaces have the same topology, then their -th homotopy groups are isomorphic for all . However, the real plane has exactly the same homotopy groups as a solitary point (as does a Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also make certain computations easier. Low-dimensional examples The low-dimensional examples of homotopy groups of spheres provide a sense of the subject, because these special cases can be visualized in ordinary 3-dimensional space. However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres. The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a rubber band around one's finger: it can be wrapped once, twice, three times and so on. The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group is therefore an infinite cyclic group, and is isomorphic to the group of integers under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the winding number of a loop around the origin in the plane. The identification (a group isomorphism) of the homotopy group with the integers is often written as an equality: thus . Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing