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Published online by Cambridge University Press: 20 January 2009
The definitions of the various proper homotopy groups correspond to three main geometrical ideas: sequences of spheres converging to a Freudenthal end (Brown groups); infinite cylinders giving the mobility of spheres towards a proper end (Čerin-Steenrod groups); sequences of spheres, each one movable to the next one following a proper end (Čech groups). The Brown and Čech groups have a rather complex structure and the calculations of these groups are very difficult (see [4]). The Čerin-Steenrod groups have a much simpler structure and this fact eases the computations.
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