Co-H-Space

A co--space is a pointed space admitting a pointed map such that collapsing either copy after applying gives a map homotopic through pointed maps to the identity map of . Here is the quotient space formed from the disjoint union of two copies of by identifying their basepoints.

This map induces a binary operation on the based homotopy classes for each pointed space . A group structure requires additional homotopy conditions corresponding to associativity and inverses. The spheres for , including the circle, have these additional structures, giving the homotopy groups .