Global Study of the Stable Homotopy Category and the Kervaire Invariant Problem

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1 For fixed prime number p and a natural number n, a natural number d (【greater than or equal】 n) has been shown to exist, satisfying the following : Those properties of the p-cpmpletion of a finite spectrum which can be "seen"by K (0), K (1), … , K (n - 1), can be "seen"by just the higher Morava K -theories K (m), K (m + 1), … , K (m + d). This may be regarded as a partial evidence of Hopkinas' chromatic splitting conjecture.2 Applying the philosophy of Hopkins ' chromatic splitting conjecture, which concerns the chromatic tower in the stable homotopy category, to the space-level unstable homotopy category, new concepts "sparce" and "mod q stupport" were introduced to generalize a theorem of Ravenel-Wilson-Yagita.3 E (n)-based modified Adams-Novikov spectral sequence via injective resolutions, which compute the abelian group [X, Y] of the set of homotopy classes between E (n)-local spectra X, Y, has been shown to posses a horizontal vanishing line, which is independent of X, Y.4 Replacing " suspension" by "join, " G-joiin Theorem, an unbased analogue of the G-Freudenthal Suspension Theorem in the equivariant homotopy theory, has been formulated and proven. This result immediately implies some Borsuk-Ulam type theorem.5 (Mostly with M.Furuta, Y.Kametani, and H.Matsue) Applying the G-join Theorem to the Stable Homotopy Seiberg-Witten Invariants, Furuta's 10/8 Theorem concerning Spin closed 4 manifolds has been improved. Also, "adjunction inequality" has been obtained for some Spin closed 4 manifolds6 (Mostly with M.Furuta and Y.Kametani) For any closed 4 manifold with b^+_2 【greater than or equal】 1, its sufficiently many iterated connected sum with itself has been shown to carry the trivial stable homotopy Seiberg-Witten invariants for any Spin^c structure, using the Devinatz-Hopkins-Smith nilpotency theorem.