The topology of the space of submanifolds and its geometric applications

研究了可数维欧氏空间的标记位形空间的同伦型，并证明了通过代数补全标记空间可以得到其同伦论群补全。在应用中，这意味着来自可加整数群的任意有限子集的同伦理论与稳定同伦理论相吻合。这是对Caruso“可数维欧几里得空间中正负粒子空间”同伦无限环空间QS^0的结果的巨大改进。我们还建立了用连续函子的概念代替谱来构造广义同调和上同调理论的方法。这使我们能够以一种简单而有效的方式统一各种（co）同源理论的定义。此外，该方法适用于理论构造的等价推广。除了上述的基础研究外，我们还将构形空间理论应用于几何问题，并获得了几个关于全纯映射空间拓扑的有趣结果。

We studied the homotopy type of the labeled configuration space of the countable dimensional Euclidean space and showed that its homology-theoretic group-completion can be obtained by algebraically completing the label space. Among applications, this implies the result that the homology theory coming from arbitrary finite subset of the additive group of integers coincides with the stable homotopy theory. This is the vast improvements of the results Caruso that the space of "positive and negative particles in the countable dimensional Euclidean space" is homotopic the infinite loop space QS^0.We also established the method for constructing generalized homology and cohomology theories by using the notion of continuous functors, instead of spectra. This enables us to uniformize the definitions of various (co)homology theories in a simple and efficient manner. Furthermore, this methods is suitable for equivariantly generalize the construction of theories.In addition to the fundamental studies stated above, we applied our theory of configuration spaces to geometric problems and obtained several interesting results concerning the topology of the space of holomorphic maps.