AI 中文总结
研究凯莱射影平面加厚\(\mathbb{OP}^{2}\times\mathbb{D}^{k}\)（\(k\geq1\)且\(k\equiv0\pmod4\)）的相对光滑手术结构集，通过确定法不变量等计算，有构造不同胚流形、计算同伦群、构造特殊丛等应用。
AI 中文摘要
我们通过确定相应的法不变量和手术障碍映射，计算了对于每个\(k\geq1\)且\(k\equiv0\pmod4\)的凯莱射影平面\(\mathbb{OP}^{2}\)加厚\(\mathbb{OP}^{2}\times\mathbb{D}^{k}\)的相对光滑手术结构集。我们证明了后者不是满射，并确定了其像的生成元的\(2\)-adic赋值。作为应用，我们构造了无限多个两两不同胚的\(16 + k\)维闭光滑流形，它们与\(\mathbb{OP}^{2}\times\mathbb{S}^{k}\)同伦等价且由其总庞特里亚金数区分；计算了在每个与\(3\)模\(4\)同余的度数下块微分同胚群\(\widetilde{\operatorname{Diff}}(\mathbb{OP}^{2})\)的有理同伦群；并构造了在\(\mathbb{S}^{4}\)、\(\mathbb{S}^{8}\)和\(\mathbb{S}^{12}\)上的光滑\(\mathbb{OP}^{2}\)-丛，其全空间具有非零的\(\widehat{\mathfrak{A}}\)-亏格。这些丛在\(\mathbb{OP}^{2}\)上正截面、里奇和标量曲率度量空间的同伦群中产生了无限阶元素。
英文摘要
We compute the relative smooth surgery structure sets of the thickenings $\mathbb{OP}^{2}\times\mathbb{D}^{k}$ of the Cayley projective plane $\mathbb{OP}^{2}$ for every $k\geq 1$ with $k\equiv 0\pmod 4$, by determining the corresponding normal invariants and surgery obstruction map. We show that the latter is not surjective and determine the $2$-adic valuation of the generator of its image. As applications, we construct infinitely many pairwise non-homeomorphic closed smooth manifolds of dimension $16+k$, homotopy equivalent to $\mathbb{OP}^{2}\times\mathbb{S}^{k}$ and distinguished by their Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group $\widetilde{\operatorname{Diff}}(\mathbb{OP}^{2})$ in every degree congruent to $3$ modulo $4$; and we construct smooth $\mathbb{OP}^{2}$-bundles over $\mathbb{S}^{4}$, $\mathbb{S}^{8}$, and $\mathbb{S}^{12}$ whose total spaces have non-vanishing $\widehat{\mathfrak{A}}$-genus. These bundles yield elements of infinite order in the homotopy groups of the spaces of metrics of positive sectional, Ricci, and scalar curvature on $\mathbb{OP}^{2}$ in degrees $3$, $7$, and $11$.
Comments31 pages, Comments are welcome
