Size homotopy group explained

(M,\varphi) is given, where M is a closed manifold of class C ^{0} and \varphi:M\to R ^{k} is a continuous function. Consider the lexicographical order \preceq on 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 R ^{k} set M _{Y} = \{Z\in 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 ≈ _{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 ≈ _{Y} , endowed with the operation induced by the usual composition of based loops.^[1] (M,\varphi) computed at (X,Y) and P is the imageh _{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.

See also

• Size function
• Size functor
• Size pair
• Natural pseudodistance

Notes and 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.