[论文解读] Notes on explicit smooth maps on 7-dimensional manifolds into the 4-dimensional Euclidean space

Naoki Kitazawa|arXiv (Cornell University)|Nov 25, 2019

Homotopy and Cohomology in Algebraic Topology参考文献 21被引用 11

一句话总结

本文構造了從7維緊緻流形到4維歐幾里得空間的顯式折疊地圖，並透過Reeb空間與上同調環分析其拓撲與幾何性質。研究確立了某些7維流形（包括異常球面）無法容許特定的通用地圖映射至R⁴，同時透過折疊地圖奇點的結構及其在上同調與同調群上誘導的映射，提供了新的代數拓撲不變量。

ABSTRACT

A fold map is a smooth map at each singular point of which it is represented as the product map of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function can be taken as a natural height function on an unit disk. The class of special generic maps includes a Morse function with exactly two singular points on a closed manifold, characterizing a sphere topologically (except 4-dimensional cases) as the Reeb's theorem shows, and canonical projections of unit spheres. It has been known that so-called exotic spheres do not admit special generic maps into Euclidean spaces whose dimensions are sufficiently high and smaller than the dimensions of the spheres. Exotic 7-dimensional homotopy spheres do not admit special generic maps into the 4-dimensional Euclidean space for example. We can easily obtain special generic maps on fundamental manifolds such as ones represented as connected sums of products of two standard spheres and in considerable cases, smooth manifolds resembling topologically them and different from them do not admit special generic maps. These interesting results are due to studies of Saeki, Sakuma and Wrazidlo since the 1990s. In the present paper, we present new results on explicit smooth maps including fold maps on 7-dimensional manifolds into the 4-dimensional Euclidean space and meanings in algebraic topology and differential topology of manifolds. Moreover, the author obtained related results before motivated by the studies before and they are reviewed in the presentation of the new results. We also present new discussions and results related to the results for 7-dimensional manifolds and maps on them for fold maps between manifolds of general dimensions.

研究动机与目标

• 構造從7維緊緻流形到4維歐幾里得空間的顯式光滑折疊地圖。
• 研究阻止異常7維同倫球面容許特殊通用地圖映射至R⁴的拓撲障礙。
• 分析容許此類折疊地圖之流形的代數拓撲性質，特別著重於上同調環與同調群。
• 將折疊地圖Reeb空間的結果擴展至流形維度相對於目標空間不夠高的情況。
• 系統性地研究由具有嵌入奇點集與標準球面纖維的折疊地圖所誘導的上同調與同調群上的映射。

提出的方法

• 使用索引為0與1的局部模型，構造一般維度流形之間的折疊地圖，其形式基於標準折疊地圖：$(x_1,\dots,x_m) \mapsto (x_1,\dots,x_{n-1}, \sum_{k=n}^{m-i} x_k^2 - \sum_{k=m-i+1}^{m} x_k^2)$。
• 對正則纖維為標準球面不交並的折疊地圖應用Reeb空間構造，且其奇點集限制為嵌入。
• 透過特徵類與杯積的拉回，分析上同調與同調群上的誘導同態。
• 利用Reeb空間的結構，將流形的上同調環與折疊地圖的像及奇點集關聯起來。
• 應用奇點理論與莫爾斯理論的已知結果，將折疊地圖分類推廣至特殊通用情形之外。
• 運用杯積關係與庞加萊對偶性等代數工具，推導出容許此類地圖之7維流形上同調環的約束。

实验结果

研究问题

• RQ1 哪些7維流形容許映射至R⁴的折疊地圖？其背後的拓撲障礙為何？
• RQ2 為何異常7維同倫球面無法容許映射至R⁴的特殊通用地圖？這與其光滑結構有何關聯？
• RQ3 7維流形的上同調環與同調群如何透過折疊地圖下的Reeb空間結構相互關聯？
• RQ4 具有嵌入奇點集與標準球面纖維的折疊地圖會保留或誘導哪些代數不變量？
• RQ5 當流形維度不夠高時，Reeb空間構造在多大程度上能恢復流形的上同調環結構？

主要发现

• 異常7維同倫球面無法容許映射至R⁴的特殊通用地圖，確認了微分拓撲中已知的障礙。
• 對於滿足奇點集嵌入且纖維微分同胚於$ S^3 $或$ S^3 \sqcup S^3 $的折疊地圖$ f: M^7 \to \mathbb{R}^4 $，Reeb空間繼承了重要的拓撲不變量。
• 7維流形$ M $的上同調環受到來自Reeb空間拉回的杯積結構的約束，特別是透過關係如$ a_j^* \cdot \phi_{A,m-k}(a_j) $生成$ H^m(M;\mathbb{Z}) $。
• 某些杯積項消失（例如$ a_j^* \cdot \phi_{n,m-n}((0,c)) = 0 $），顯示折疊地圖誘導的上同調環中存在非平凡關係。
• 乘積$ b_{j,n}^* \cdot \phi_{n,m-n}((b_j,0)) $構成$ H^m(M;\mathbb{Z}) $的生成元，顯示特定上同調類與奇點集結構密切相關。
• 映射$ f|_{S(f)} $為嵌入，且所有奇點的指數為0或1，確保Reeb空間性質良好，並允許對拓撲進行代數控制。