Homotopy Sets and Groups

Abstract

The functors F _K and cofunctors F ^‪ _K introduced in 1.6.iii) are especially important in algebraic topology. In this chapter we give conditions on (K, k₀) which guarantee that F _K (X, x₀) and F ^‪ _K (X, x₀) are groups for all pointed spaces (X, x₀)—i.e. conditions on (K, k₀) which make F _K (resp. F ^‪ _K ) a functor (resp. cofunctor) from Pℐ (or P ℐ) to c. In particular, if ${S^{n}} = \left\{ {x \in {\mathbb{R}^{{n + 1}}}:\left\| x \right\| = 1} \right\}$ is a pointed space such that ${F_{{{s^{n}}}}}:PT' \to G$ and we investigate some of the properties of ${\pi _{n}}\left( {X,{x_{0}}} \right) = {F_{{{s^{n}}}}}\left( {X,{x_{0}}} \right),n \geqslant 0.$.