Topology related to Mathematical Physics, Morse Theory and Numerical Computations

在此之前，M.Guest教授、A.Kozlowski教授和作者证明了Atiyah-Jones-Segal型定理对从1维复射影空间到某些复射影簇族的全纯映射空间成立。现在，他表明，一个类似的结果持有某些子空间，他们的定义是使用的概念，多重诱导的陈述多项式的全纯映射。到目前为止，我们通常研究的是AJS型定理对真实的维（或复的一维）空间上的全纯（或代数）映射空间是否成立。在我们的研究中，现在我们可以研究这样的问题是否从高维空间的全纯或代数映射的空间。作为一个例子，我们证明了从某些紧仿射空间到复或真实的Grassmanian流形的正则映射空间是这些空间之间的连续映射空间的同伦等价，如果这些簇的仿射空间满足向量丛的某些条件，这是与A.科兹洛夫斯基为了证明这些结果，我们使用了真实的代数几何的技巧。此外，我们可以证明，AJS型定理成立的空间，利用上述定理。特别地，当m=n-1或m=n时，我们还确定了从m维真实的射影空间到n维射影空间的映射空间的基本群.这一结果可以看作是上述第一种情况研究的真实的版本。我们还从奇点理论的新观点出发，利用分割理论研究了例外手术。特别是，我们研究了这种手术的机制和特殊手术的结构。

Previously, Professors M.Guest, A.Kozlowski and the author showed that the Atiyah-Jones-Segal type Theorem holds for spaces of holomorphic maps from the 1 dimensional complex projective space to certain family of complex projective varieties. Now he showed that a similar result holds for certain subspaces of them which are defined by using the concept of multiplicities induced from the representations of polynomials of holomorphic maps. Furthermore, he computed the fundamental groups for spaces of self-holomorphic maps on the n dimensional complex Projective spaces.Until now, we usually investigate whether AJS type Theorem holds or not for spaces of holomorphic (or algebraic) maps from one real dimensional (or complex one dimensional) spaces. In our investigation, now we can investigate whether such a problem for spaces of holomorphic or algebraic maps from high dimensional spaces. As one example, we can show that the spaces of regular maps from certain compact affine spaces into complex or real Grassmanian manifolds are homotopy equivalent of spaces of continuous maps between these spaces if these varieties Affine spaces satisfy certain conditions of vector bundles, which is one of joint works with Professor A. Kozlowski. To prove these results, we use the technique of real algebraic geometry. Moreover, we can prove that AJS type Theorem holds for such spaces by using the above Theorem. In particular, we also determine the fundamental groups of spaces of maps from m dimensional real projective space into n dimensional one when m=n-1, or m=n. Such a result can be regarded as a real version of the study investigated in the above first case.We also study the exceptional surgery from the new point view of singularity theory by using the divide theory. In particular, we study the mechanism of such surgeries and the structure of the set of exceptional surgeries.