On stable homotopy of the complex projective space

Hypersurface representation and the image of the double S³–transfer

“…Mukai [8,Theorem 2] has shown that both ν * and η * are in the image of the S 1 -transfer homomorphism τ : π s i−1 (CP ∞ ) → π s i (S 0 ). Morisugi [7, Corollary E] has shown that all Mahowald's elements are in the image of the S 3 -transfer homomorphism π s * (Q ∞ ) → π s * (S 0 ) given for the quaternionic quasi-projective space Q ∞ = n Q n .…”

Two-primary algebraic K-theory of pointed spaces

“…This is the wedge sum of the (desuspended) S 1 -transfer map CP ∞ → S −1 , and the (desuspended) multiplication by η map S 0 → S −1 . The image of the S 1transfer map was computed in dimensions * ≤ 20 by Mukai in [Mu1], [Mu2] and [Mu3], and we use these results to determine the differentials in (2.1) landing in filtration s = −1 in the same range of dimensions.…”

“…The differentials landing in filtration s = 0 are always zero, as noted above. The differentials landing in filtration s = −1 are determined by the computation of the S 1 -transfer in [Mu1] and [Mu2]. Thus…”

“…The E 1 -term in this spectral sequence is given in terms of the stable homotopy groups of spheres, π S * , and the differentials depend on the attaching maps for the cells in CP ∞ −1 . This involves primary and secondary operations in homotopy, somewhat along the lines of Toda's book [To], and we build on previous work for CP ∞ by Mosher [Mo] and Mukai [Mu1], [Mu2] and [Mu3].…”

Algebraic 𝐾-Theory

“…Hence the E 2 -term of this spectral sequence has a copy of the stable homotopy groups of spheres (the stable stems) π S * = π * QS 0 , in each even column starting in filtration degree −2. Based on work by Mosher [Mo] and Mukai [Mu1,Mu2,Mu3], the author as made such calculations in the range of total degrees ≤ 20, where there are approximately 100 nonzero differentials.…”