Understanding exotic spheres from the viewpoint of global singularity theory of smooth maps

在最近的预印本(arxiv：http://arxiv.org/abs/2009.05928)，)中，我们研究了有理同调球面的特殊一般映射的存在性和构造问题。我们方法的新奇之处在于考虑了这类流形的积分同调的扭子群。证明了：如果奇数维n=2k+1&gt；4的有理同调球面允许一个特殊的一般映射进入&lt；n维欧氏空间，则它的k次积分同调群的基数是平方数.一方面，我们证明了在我们的同调条件下，任何平方都可以实现。另一方面，也有一些有理同调球面的例子不满足我们的同调条件。我们的结果为后面的项目铺平了道路，在该项目中，我们研究了高连通流形的特殊一般映射的链接形式，它是交集形式的扭转模拟。在另一个项目中，我们开发了一种基于Sullivan的PL多项式微分形式的交集空间的新方法。我们的结果暗示了交空间的有理上同调环的唯一性。这一结果是该研究领域的一个新发现，我们发表了孤立奇点的情况。在正在进行的工作中，我们沿着Agustin和Fernandez de Bobadilla的构造将我们的方法推广到一类具有任意分层深度的奇异空间，包括环簇。此外，在与T.Essig的合作中，我们正在完成一份关于构造二层深度交集空间的基本类的预印本。

In a recent preprint (arXiv: http://arxiv.org/abs/2009.05928), we studied the existence and construction problems for special generic maps of rational homology spheres. The novelty of our approach is to consider the torsion subgroup of the integral homology of such manifolds. We showed that if a rational homology sphere of odd dimension n = 2k + 1 > 4 admits a special generic map into a Euclidean space of dimension < n, then the cardinality of its integral homology group of degree k is a square. On the one hand, we showed that any square can can be realized in our homological condition. On the other hand, there are examples of rational homology spheres that do not satisfy our homological condition. Our results paved the way to a subsequent project, in which we study special generic maps of highly connected manifolds in terms of the linking form, which is a torsion analog of the intersection form. In another project, we developed a new approach to intersection spaces that is based on Sullivan's PL polynomial differential forms. Our result implies uniqueness of the rational cohomology ring of intersection spaces. This result is a new discovery in the research field, and we published the case of isolated singularities. In ongoing work, we generalize our approach along the construction of Agustin and Fernandez de Bobadilla to a class of singular spaces of arbitrary stratification depth including toric varieties. Moreover, in joint work with T. Essig, we are finalizing a preprint about the construction of a fundamental class for intersection spaces in stratification depth two.