- Size homotopy group
The concept of size homotopy group is the anologous in
size theory of the classical concept ofhomotopy group . In order to give its definition, let us assume that asize pair M,varphi) is given, where M is aclosed manifold of class C^0 and varphi:M o mathbb{R}^k is acontinuous function . Let us consider thepartial 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 Yinmathbb{R}^k we set M_{Y}={Zinmathbb{R}^k:Zpreceq Y} .Assume that Pin M_X and Xpreceq Y . If alpha , eta are two paths from P to P and a
homotopy from alpha to eta , based at P , exists in thetopological space M_{Y} , then we write alpha approx_{Y}eta . The first size homotopy group of thesize pair M,varphi) computed at X,Y) is defined to be thequotient set of the set of allpath s from P to P in M_X with respect to theequivalence relation approx_{Y} , endowed with the operation induced by the usual composition of basedloop s 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 .] .In other words, the first size homotopy group of the
size pair M,varphi) computed at X,Y) and P is the imageh_{XY}(pi_1(M_X,P)) of the firsthomotopy group pi_1(M_X,P) with base point P of thetopological space M_X , when h_{XY} is thehomomorphism induced by the inclusion of M_X in M_Y .The n -th size homotopy group is obtained by substituting the
loop s based at P with thecontinuous function s alpha:S^n o M taking a fixed point of S^n to P , as happens when higherhomotopy group s are defined.References
ee also
*
Size theory
*Size function
*Size functor
*Size pair
*Natural pseudodistance
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