An example for homotopy groups with coefficients

Abstract

In this note we present two spaces X and Y all of whose homotopy groups are isomorphic, but whose homotopy groups with coefficients are not isomorphic for a certain coefficient group.' This example depends on the fact that the universal coefficient sequence [2], which relates the ordinary homotopy groups to those with coefficients, does not split. The spaces X and Y will be 1-connected, CW-complexes and the coefficient group will be Zm, the integers modulo m. We adopt the following notation: M(G, p) denotes a Moore complex of type (G, p) (i.e., a space with a single nonvanishing homology group G in dimension p) and K(G, p) denotes an Eilenberg-MacLane complex of type (G, p) (i.e., a space with a single nonvanishing homotopy group G in dimension p). Recall that 7rr(G; A), the rth homotopy group of the space A with coefficients in the group G, is the group of homotopy classes of base point preserving maps from M(G, r) into A. If Z is the group of integers, then 7r(Z; A) =irr(A), the rth homotopy group of A. Finally we recall the universal coefficient theorem [2] which asserts the exactness of the following sequence