Size homotopy group

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 ) {\displaystyle (M,\varphi )} is given, where M {\displaystyle M} is a closed manifold of class C 0 {\displaystyle C^{0}\ } and φ : M → R k {\displaystyle \varphi :M\to \mathbb {R} ^{k}} is a continuous function. Consider the lexicographical order ⪯ {\displaystyle \preceq } on R k {\displaystyle \mathbb {R} ^{k}} defined by setting ( x 1 , … , x k ) ⪯ ( y 1 , … , y k ) {\displaystyle (x_{1},\ldots ,x_{k})\preceq (y_{1},\ldots ,y_{k})\ } if and only if x 1 ≤ y 1 , … , x k ≤ y k {\displaystyle x_{1}\leq y_{1},\ldots ,x_{k}\leq y_{k}} . For every Y ∈ R k {\displaystyle Y\in \mathbb {R} ^{k}} set M Y = { Z ∈ R k : Z ⪯ Y } {\displaystyle M_{Y}=\{Z\in \mathbb {R} ^{k}:Z\preceq Y\}} . Assume that P ∈ M X {\displaystyle P\in M_{X}\ } and X ⪯ Y {\displaystyle X\preceq Y\ } . If α {\displaystyle \alpha \ } , β {\displaystyle \beta \ } are two paths from P {\displaystyle P\ } to P {\displaystyle P\ } and a homotopy from α {\displaystyle \alpha \ } to β {\displaystyle \beta \ } , based at P {\displaystyle P\ } , exists in the topological space M Y {\displaystyle M_{Y}\ } , then we write α ≈ Y β {\displaystyle \alpha \approx _{Y}\beta \ } . The first size homotopy group of the size pair ( M , φ ) {\displaystyle (M,\varphi )\ } computed at ( X , Y ) {\displaystyle (X,Y)\ } is defined to be the quotient set of the set of all paths from P {\displaystyle P\ } to P {\displaystyle P\ } in M X {\displaystyle M_{X}\ } with respect to the equivalence relation ≈ Y {\displaystyle \approx _{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 ) {\displaystyle (M,\varphi )\ } computed at ( X , Y ) {\displaystyle (X,Y)\ } and P {\displaystyle P\ } is the image

h X Y ( π 1 ( M X , P ) ) {\displaystyle h_{XY}(\pi _{1}(M_{X},P))\ }

of the first homotopy group π 1 ( M X , P ) {\displaystyle \pi _{1}(M_{X},P)\ } with base point P {\displaystyle P\ } of the topological space M X {\displaystyle M_{X}\ } , when h X Y {\displaystyle h_{XY}\ } is the homomorphism induced by the inclusion of M X {\displaystyle M_{X}\ } in M Y {\displaystyle M_{Y}\ } . The n {\displaystyle n} -th size homotopy group is obtained by substituting the loops based at P {\displaystyle P\ } with the continuous functions α : S n → M {\displaystyle \alpha :S^{n}\to M\ } taking a fixed point of S n {\displaystyle S^{n}\ } to P {\displaystyle P\ } , as happens when higher homotopy groups are defined.

Tags