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
, φ )
is a closed manifold of class
is a continuous function.
Consider the lexicographical order
defined by setting
set
Assume that
α
are two paths from
α
, based at
, exists in the topological space
, then we write
The first size homotopy group of the size pair
is defined to be the quotient set of the set of all paths from
with respect to the equivalence relation
, endowed with the operation induced by the usual composition of based loops.
[1] In other words, the first size homotopy group of the size pair
of the first homotopy group
with base point
of the topological space
is the homomorphism induced by the inclusion of
-th size homotopy group is obtained by substituting the loops based at
with the continuous functions
taking a fixed point of
, as happens when higher homotopy groups are defined.
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