发表机构
Middle East Technical University（中东理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将 Hambleton-Kreck 的交换辫图方法推广到高维流形，计算球面乘积与连通和的同伦自等价群，得到短正合序列并揭示同伦与微分同胚分类的差异。
AI 中文摘要
设 $M$ 为闭定向流形。基于自等价群 $\operatorname{Aut}_\bullet(M)$ 是流形分类中的一个自然不变量，Hambleton 和 Kreck 的交换辫图给出了计算 $4$-流形该群的系统方法。我们将他们的框架推广到维数为 $2n$ 或 $2n-1$ 的 $BO\langle n+1\rangle$-流形，在以下两个假设下：(F) 结构映射通过 Postnikov $n$-型分解，(T) 相关单值性平凡。当 $M$ 稳定可平行化时，这两个假设均成立，而我们所处理的每个例子都属于这种情况。随后我们讨论球面的乘积与连通和。对于 $S^n\times S^n$，相关相交形式的等距群在 $n$ 为奇数时是辛群 $Sp_2(\mathbb{Z})=SL_2(\mathbb{Z})$，在 $n$ 为偶数时是有限群 $(\mathbb{Z}/2)^2$，我们通过 Whitehead 积论证直接验证了这一点。将其代入辫图，在每种情况下都得到一个短正合序列 $0\to K\to\operatorname{Aut}_\bullet(M)\to Q\to0$，其中 $Q$ 是上述等距群，而 $K$ 是由球面的不稳定同伦群构造的配边理论核。我们对 $S^1\times S^{m-1}$ 进行了详细计算；对 $S^3\times S^3$，序列分裂为直积；对 $r$ 个 $S^3\times S^3$ 的连通和，置换被加项对 $K$ 的作用非平凡；对 $S^3\times S^4$；以及对 $S^4\times S^4$，其中 $Q$ 完全确定但 $K$ 仅为猜想。我们还讨论了连通和 $(S^3\times S^5) \connsum (S^3\times S^5)$，同样的方法虽未能完全闭合但已接近答案。全文通过与光滑结构集的比较，展示了同伦分类与微分同胚分类可能具有非常不同的行为。
英文摘要
Let $M$ be a closed, oriented manifold. The group $\operatorname{Aut}_\bullet(M)$ of based self-equivalences of $M$ is a natural invariant in manifold classification, and Hambleton and Kreck's commutative braid gives a systematic way to compute it for $4$-manifolds. We extend their framework to $BO\langle n+1\rangle$-manifolds of dimension $2n$ or $2n-1$, under two hypotheses: (F), that the structure map factors through the Postnikov $n$-type, and (T), triviality of the relevant monodromy. Both hold whenever $M$ is stably parallelizable, which is the case in every example we treat. We then discuss products and connected sums of spheres. For $S^n\times S^n$, the isometry group of the relevant intersection form is the symplectic group $Sp_2(\mathbb{Z})=SL_2(\mathbb{Z})$ when $n$ is odd, and the finite group $(\mathbb{Z}/2)^2$ when $n$ is even, which we verify directly with a Whitehead-product argument. Feeding this into the braid gives, in each case, a short exact sequence $0\to K\to\operatorname{Aut}_\bullet(M)\to Q\to0$, with $Q$ this same isometry group and $K$ a bordism-theoretic kernel built from unstable homotopy groups of spheres. We work this out for $S^1\times S^{m-1}$; for $S^3\times S^3$, where the sequence splits as a direct product; for connected sums of $r$ copies of $S^3\times S^3$, where permuting the summands acts non-trivially on $K$ instead; for $S^3\times S^4$; and for $S^4\times S^4$, where $Q$ is fully determined but $K$ is only conjectured. We also discuss the connected sum $(S^3\times S^5)$ \connsum $(S^3\times S^5)$, where the same method gets most of the way to an answer without quite closing it. Throughout, comparing with the smooth structure set shows how differently homotopy and diffeomorphism classification can behave.
Comments34 pages
