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

稳定同伦范畴和Kervaire不变问题的全局研究

基本信息

批准号：
11640072
负责人：
MINAMI Norihiko
金额：
$ 2.18万
依托单位：
Nagoya Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640072/
关键词：
Stable Homotopy Category Kervaire Invariant Problem K-Theory Chromatic Tower Equivariant Homotopy Theory 4 dimensional Manifolds Seiberg-Witten Invariants Adjunction Inequality 安定ホモトピー論 BP理論 Adamsスペクトラル系列 Nilpotency 安定ホモトピーSeiberg-

项目摘要

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.

1 对于固定素数 p 和自然数 n，已证明存在一个自然数 d（【大于或等于】 n），满足以下条件：有限谱的 p-c 完成的那些性质可以被 K (0), K (1), … , K (n - 1) “看到”，可以通过更高的 Morava K 理论 K (m), K (m + 1), … , K (m) “看到” + d).这可以看作是Hopkinas色分裂猜想的部分证据。2将Hopkins色分裂猜想中稳定同伦范畴中的色塔的哲学应用到空间级不稳定同伦范畴，引入新概念“sparce”和“mod q stupport”，推广了Ravenel-Wilson-Yagita定理。 3 E 基于 (n) 的修改 Adams-Novikov 谱序列通过单射分辨率计算 E (n)-局部谱 X, Y 之间同伦类集合的阿贝尔群 [X, Y]，已被证明拥有一条水平消失线，该线与 X, Y 无关。4 用“连接”代替“悬浮”，G-连接定理是等变同伦理论中的G-弗洛伊登塔尔悬浮定理已被表述并证明。这个结果立即暗示了一些 Borsuk-Ulam 型定理。5（主要是 M.Furuta、Y.Kametani 和 H.Matsue）将 G-join 定理应用于稳定同伦 Seiberg-Witten 不变量，改进了关于自旋闭 4 流形的 Furuta 10/8 定理。此外，对于某些自旋闭 4 流形6（主要是 M.Furuta 和 Y.Kametani），已经获得了“附加不等式”。对于任何 b^+_2 【大于或等于】1 的闭 4 流形，其足够多的迭代连通和已被证明可以携带任何 Spin^c 结构的平凡稳定同伦 Seiberg-Witten 不变量，使用 Devinatz-Hopkins-Smith 幂零定理。