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. Let us consider the partial order \preceq\ in \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\ we 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.[1]

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.

References

  1. Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464, 1999.

See also