Cohomotopy sets of projective planes

概要

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We set F=R(real), C(complex), H(quaternion), O(octonian) and d=dimRF. We denote by FP² the F-projective plane. The purpose of this note is to determine the cohomotopy set πⁿ(FP²) = [FP², Sⁿ]. Let h=h(F): S²d⁻¹→Sd be the Hopf map. Then we have a cell structure FP²=SdUhℯ²d and a cofiber sequence: S²d⁻¹ h→Sd- i→FP² p→S²d ∑h→Sd⁺¹→..., (1) where i is the inclusion map, p = p(F) is a map pinching Sd to one point and ∑h is the reduced suspension of h. Our result is given by the table on page 7. Its essence is stated as follows.