Bauer-Furuta族不变量、奇异曲面与Smale猜想
Family Bauer--Furuta invariant, Exotic Surfaces and Smale conjecture
摘要：
我们建立了一对奇异曲面的存在性，在穿孔$K3$保持异国情调后，一个外部稳定化，并有同构补。证明中的一个关键要素是一个消灭定理的家庭Bauer-古田不变量的一个大家庭的自旋4-流形上，这是证明使用汤姆Dieck分裂定理在等变稳定同伦理论。特别地，我们证明了$S^{4}$上的任何保向同调的$S^{1}$ -等变族Bauer-Furuta不变量是平凡的，并且当同调作用于$S^{2}\times S^{2}$上的同调时，$\mathrm{Pin}(2)$ -等变族Bauer-Furuta不变量是平凡的.因此，这些不变量不能检测$S^{4}$或$S^{2}\times S^{2}$上的奇异自同构。此外，我们的定理也适用于某些奇异的循环上的$S^{4}$（最近发现的渡边），并表明这些循环有平凡的家庭鲍尔-古田不变量。在途中，我们观察到一个奇怪的元素在$\mathrm{Pin}(2)$ -等变稳定同伦群的领域，它可能被用来检测一个奇异的同形$S^{4}$。
英文摘要：
We establish the existence of a pair of exotic surfaces in a punctured $K3$ which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer--Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the $S^{1}$-equivariant family Bauer--Furuta invariant of any orientation-preserving diffeomorphism on $S^{4}$ is trivial and that the $\mathrm{Pin}(2)$-equivariant family Bauer--Furuta invariant for a diffeomorphism on $S^{2}\times S^{2}$ is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic self-diffeomorphisms on $S^{4}$ or $S^{2}\times S^{2}$. Furthermore, our theorem also applies to certain exotic loops of diffeomorphisms on $S^{4}$ (as recently discovered by Watanabe) and show that these loops have trivial family Bauer--Furuta invariants. En route, we observe a curious element in the $\mathrm{Pin}(2)$-equivariant stable homotopy group of spheres which could potentially be used to detect an exotic diffeomorphism on $S^{4}$.
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