The spaces of regular maps and the applications of real singularity theory to homotopy theory

本文的主要目的是探讨G. Segal （Acta mathematical）对有理函数空间功的推广。, 1979)。本文研究了映射空间Map（M，N）在M和N为dim M >1流形时的拓扑结构，主要研究了M和N为实射影空间的情况。在这种情况下，我们尝试用多项式表示的代数映射（或正则映射）的有限维子空间Alg（M，N）来近似它。特别地，我们可以通过与A. Kozlowski（华沙大学）的联合工作确定它的近似同伦型的显维数。

The main purpose of this research is to investigate the generalization of the work of spaces of rational functions due to G. Segal (Acta Math., 1979). In this research, we consider the topology of the mapping space Map(M,N) when M and N are manifolds with dim M >1, and we mainly study the case that M and N are real projective spaces. In this case, we try to approximate it by a finite dimensional subspace Alg(M,N) of algebraic maps (or regular maps) which are represented by polynomials.In particular, we can determine the explicit dimension of approximating homotopy type of it by the joint work with A. Kozlowski (Warsaw University) .