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

1对于固定素数p和自然数n，证明了自然数d（大于或等于）n的存在性，满足以下条件：有限谱的p-完备性可以被K (0)， K(1)，…，K （n - 1）“看到”的性质，只能被更高的Morava K -理论K (m), K (m + 1)，…，K （m + d）“看到”。这可以看作是Hopkinas色分裂猜想的部分证据将霍普金斯关于稳定同伦范畴中的色塔的色分裂猜想的哲学应用到空间层面的不稳定同伦范畴，引入了“空间”和“模q stupport”这两个新概念，推广了ravenel - wilson - yagita的一个定理基于E (n)的改进Adams-Novikov谱序列通过内射分解，计算E (n)-局部谱X， Y之间的阿贝耳群[X， Y]，已经证明具有一条水平消失线，该消失线与X， Y无关。4用“连接”代替“悬挂”，g -连接定理是等价同伦理论中G-Freudenthal悬架定理的无基类比，已被表述并证明。这个结果立即暗示了一些Borsuk-Ulam型定理将G-join定理应用于稳定同伦Seiberg-Witten不变量，改进了Furuta关于自旋闭4流形的10/8定理。对于任意具有b^+_2【大于或等于】1的闭4流形，利用Devinatz-Hopkins-Smith零幂定理，证明了其与自身的足够多次迭代连通和携带任意自旋^c结构的平凡稳定同伦Seiberg-Witten不变量。

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.