Homotopy theory MOC

Homotopy group

Given a based space and , we define an -dimensional loop with base to be a continuous function , i.e. mapping the boundary of the unit hypercube to the basepoint. Given loops we define their concatenation as

The th homotopy group is the set of homotopy classes of such loops with the concatenation operation, i.e. as a set . #m/def/homotopy Equivalently, .

Properties

• gives the fundamental group
• Non-fundamental homotopy groups are abelian
• Non-fundamental homotopy groups of a circle are trivial

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