论文标题
罗奇林定理的相对版本
A relative version of Rochlin's theorem
论文作者
论文摘要
罗奇林（Rochlin）证明了一个封闭的4维连接的平稳折叠$ x^4 $，而第二个stiefel-whitney级别的消失了签名$σ（x）$ 16。 \ Mathbb {Z}/2 \ Mathbb {z}）$，dual dual dual to $ w_2（x）$，如果$ x $中的嵌入式球体可以表示$ x $，那么自我交换号$ $ξ^2 $可在16中被释放。松本。我们进一步概括了有关具有边界的4个manifolds的结果。鉴于具有整体同源球边界的平滑紧凑型四个流形$ x^4 $，并且具有连接的特征表面具有连接的边界$ f^2 $适当嵌入$ x $中，我们证明了一个定理，与$ \ f $ $ f $的ARF不变性，以及$ f $ $ f $ $ f $ $ $ $ $ $ $ $ $ $ $ $。 We then proceed to generalize this result to the case where $X$ is a topological compact orientable 4-manifold (which brings in the Kirby-Siebenmann invariant), $\partial F$ is not connected (which brings in the condition of being proper as a link), $F$ is not orientable (which brings in Brown invariants), and finally where $\partial X$ is an arbitrary 3-manifold (which brings在针结构中）。最终结果给出了对柯比·塞本曼（Kirby-Siebenmann）的“组合”描述，即具有非空边界的紧凑定位的4个manifold。
Rochlin proved that a closed 4-dimensional connected smooth oriented manifold $X^4$ with vanishing second Stiefel-Whitney class has signature $σ(X)$ divisible by 16. This was generalized by Kervaire and Milnor to the statement that if $ξ\in H_2(X;\mathbb{Z})$ is an integral lift of an element in $H_2(X; \mathbb{Z}/2\mathbb{Z})$ that is dual to $w_2(X)$, and if $ξ$ can be represented by an embedded sphere in $X$, then the self-intersection number $ξ^2$ is divisible by 16. This was subsequently generalized further by Rochlin and various alternative proofs of this result where given by Freedman, Kirby, and Matsumoto. We give further generalizations of this result concerning 4-manifolds with boundary. Given a smooth compact orientable four manifold $X^4$ with integral homology sphere boundary and a connected orientable characteristic surface with connected boundary $F^2$ properly embedded in $X$, we prove a theorem relating the Arf invariant of $\partial F$, and the Arf invariant of $F$, and the Rochlin invariant of $\partial X$. We then proceed to generalize this result to the case where $X$ is a topological compact orientable 4-manifold (which brings in the Kirby-Siebenmann invariant), $\partial F$ is not connected (which brings in the condition of being proper as a link), $F$ is not orientable (which brings in Brown invariants), and finally where $\partial X$ is an arbitrary 3-manifold (which brings in pin structures). The final result gives a "combinatorial" description of the Kirby-Siebenmann invariant of a compact orientable 4-manifold with nonempty boundary.
