The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair ( M , φ ) is given, where M is a closed manifold of class C 0 and φ : M → R k is a continuous function. Let us consider the partial order ⪯ in R k defined by setting ( x 1 , … , x k ) ⪯ ( y 1 , … , y k ) if and only if x 1 ≤ y 1 , … , x k ≤ y k . For every Y ∈ R k we set M Y = { Z ∈ R k : Z ⪯ Y } .
Assume that P ∈ M X and X ⪯ Y . If α , β are two paths from P to P and a homotopy from α to β , based at P , exists in the topological space M Y , then we write α ≈ Y β . The first size homotopy group of the size pair ( M , φ ) computed at ( X , Y ) is defined to be the quotient set of the set of all paths from P to P in M X with respect to the equivalence relation ≈ Y , endowed with the operation induced by the usual composition of based loops.
In other words, the first size homotopy group of the size pair ( M , φ ) computed at ( X , Y ) and P is the image h X Y ( π 1 ( M X , P ) ) of the first homotopy group π 1 ( M X , P ) with base point P of the topological space M X , when h X Y is the homomorphism induced by the inclusion of M X in M Y .
The n -th size homotopy group is obtained by substituting the loops based at P with the continuous functions α : S n → M taking a fixed point of S n to P , as happens when higher homotopy groups are defined.
