4维完全交的光滑分类
The smooth classification of 4-dimensional complete intersections
AI总结：
本文证明了4维完全交微分同胚分类的Sullivan猜想，通过引入Hambleton-Madsen正规映射理论等新方法，并揭示了其与$S^1$的Segal猜想之间的意外联系。
摘要：
我们证明了关于4维完全交在微分同胚意义下分类的“Sullivan猜想”。这里一个$n$维完全交是一个光滑复代数簇，由$CP^{n+k}$中$k$个超曲面的横截相交所形成。此前Kreck和Traving在64整除总次数（即定义超曲面的次数之积）的情形下证明了4维Sullivan猜想，而Fang和Klaus证明了该猜想在同伦8维球面群$Θ_8 = Z/2$的作用意义下成立。我们的证明涉及若干新思想，包括使用Hambleton-Madsen的$d$次正规映射理论，这为所有维数下的Sullivan猜想提供了新的视角。由此引出了$S^1$的Segal猜想与Sullivan猜想之间一个出人意料的联系。
英文摘要：
We prove the "Sullivan Conjecture" on the classification of 4-dimensional complete intersections up to diffeomorphism. Here an $n$-dimensional complete intersection is a smooth complex variety formed by the transverse intersection of $k$ hypersurfaces in $CP^{n+k}$. Previously Kreck and Traving proved the 4-dimensional Sullivan Conjecture when 64 divides the total degree (the product of the degrees of the defining hypersurfaces) and Fang and Klaus proved that the conjecture holds up to the action of the group of homotopy 8-spheres $Θ_8 = Z/2$. Our proof involves several new ideas, including the use of the Hambleton-Madsen theory of degree-$d$ normal maps, which provide a fresh perspective on the Sullivan Conjecture in all dimensions. This leads to an unexpected connection between the Segal Conjecture for $S^1$ and the Sullivan Conjecture.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
