HOMOTOPY PROPERTIES OF map(ΣnCP2; Sm)
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For given spaces X and Y, let map(X, Y) and map*(X, Y) be the unbased and based mapping spaces from X to Y, equipped with compact-open topology respectively. Then let map(X, Y; f) and map*(X, Y; g) be the path component of map(X, Y) containing f and map*(X, Y) containing g, respectively. In this paper, we compute cohomotopy groups of suspended complex plane πn+m(ΣnCP2) for m = 6, 7. Using these results, we classify path components of the spaces map(ΣnCP2, Sm) up to homotopy equivalence. We also determine the generalized Gottlieb groups Gn(CP2, Sm). Finally, we compute homotopy groups of map- ping spaces map(ΣnCP2, Sm; f) for all generators [f] of [ΣnCP2, Sm], and Gottlieb groups of mapping components containing constant map map(ΣnCP2, Sm; *).
Abstract
For given spaces X and Y, let map(X, Y) and map*(X, Y) be the unbased and based mapping spaces from X to Y, equipped with compact-open topology respectively. Then let map(X, Y; f) and map*(X, Y; g) be the path component of map(X, Y) containing f and map*(X, Y) containing g, respectively. In this paper, we compute cohomotopy groups of suspended complex plane πn+m(ΣnCP2) for m = 6, 7. Using these results, we classify path components of the spaces map(ΣnCP2, Sm) up to homotopy equivalence. We also determine the generalized Gottlieb groups Gn(CP2, Sm). Finally, we compute homotopy groups of map- ping spaces map(ΣnCP2, Sm; f) for all generators [f] of [ΣnCP2, Sm], and Gottlieb groups of mapping components containing constant map map(ΣnCP2, Sm; *).
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