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 $$(M,\varphi)$$ is given, where $$M$$ is a closed manifold of class $$C^0\ $$ and $$\varphi:M\to \mathbb{R}^k$$ is a continuous function. Consider the lexicographical order $$\preceq$$ on $$\mathbb{R}^k$$ defined by setting $$(x_1,\ldots,x_k)\preceq(y_1,\ldots,y_k)\ $$ if and only if $$x_1 \le y_1,\ldots, x_k \le y_k$$. For every $$Y\in\mathbb{R}^k$$ set $$M_{Y}=\{Z\in\mathbb{R}^k:Z\preceq Y\}$$.

Assume that $$P\in M_X\ $$ and $$X\preceq Y\ $$. If $$\alpha\ $$, $$\beta\ $$ are two paths from $$P\ $$ to $$P\ $$ and a homotopy from $$\alpha\ $$ to $$\beta\ $$, based at $$P\ $$, exists in the topological space $$M_{Y}\ $$, then we write $$\alpha \approx_{Y}\beta\ $$. The first size homotopy group of the size pair $$(M,\varphi)\ $$ 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 $$\approx_{Y}\ $$, endowed with the operation induced by the usual composition of based loops.

In other words, the first size homotopy group of the size pair $$(M,\varphi)\ $$ computed at $$(X,Y)\ $$ and $$P\ $$ is the image $$h_{XY}(\pi_1(M_X,P))\ $$ of the first homotopy group $$\pi_1(M_X,P)\ $$ with base point $$P\ $$ of the topological space $$M_X\ $$, when $$h_{XY}\ $$ 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 $$\alpha:S^n\to M\ $$ taking a fixed point of $$S^n\ $$ to $$P\ $$, as happens when higher homotopy groups are defined.