论文标题
旋转4个manifolds稳定差异性的代数标准
Algebraic criteria for stable diffeomorphism of spin 4-manifolds
论文作者
论文摘要
我们研究封闭，连接的，旋转4个manifolds，以通过$ s^2 \ times s^2 $的连接总和进行稳定。对于固定的基本组，有主要，次要和第三级障碍物，与签名一起导致完全稳定的分类。主要障碍物完全检测到$ \ mathbb {cp}^2 $ - 稳定的差异性，并且以前与克雷克（Kreck）和作者的代数不变性有关。 In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $τ$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs. 我们证明了我们对以下基本群体的猜想：最多3个共同体学维度的组，右角的Artin组，Abelian群体和具有季节或Abelian 2-Sylow亚组的有限群体。我们将理论运用以提供完整的代数稳定分类，以旋转$ 4 $ - manifolds，带有基本组$ \ mathbb {z} \ times \ times \ mathbb {z}/2 $。
We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature lead to a complete stable classification. The primary obstruction exactly detects $\mathbb{CP}^2$-stable diffeomorphism and was previously related to algebraic invariants by Kreck and the authors. In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $τ$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs. We prove our conjectures for the following classes of fundamental groups: groups of cohomological dimension at most 3, right-angled Artin groups, abelian groups, and finite groups with quaternion or abelian 2-Sylow subgroups. We apply our theory to give a complete algebraic stable classification of spin $4$-manifolds with fundamental group $\mathbb{Z} \times \mathbb{Z}/2$.
