On Homotopy Morphism And Its Applications

Homeomorphic morphism and homotopy equivalence are two important concepts in the theory of algebraic topology. The homotopy regular morphism has been defined in the category of topological space with base point in [22], and its existence conditions, properties and the close relationships to homotopy monomor-phism (epimorphism), homotopy regular monomorphism (epimorphism) and homotopy equivalence have also been discussed. They generalized some results of homeomorphic morphism and homotopy equivalence. In this paper, we further discussed the homotopy regular morphism and its applications.In Chapter 1, we discussed the homotopy regular morphism. It was showed at first that the loop space functor and the suspension functor preserve the properties of homotopy regular. Second, we introduced the concepts of (i,p)- homotopy inverse and group homotopy inverse respectively. Their existence and properties, have also been presented. In the last section of this chapter we introduced homology monomorphism, homology epimorphism, homology regular morphism etc. in the category of topological spaces with base point by using homology functor, discussed some properties of homology regular morphism its close relationships to homology monomorphism (epimorphism) and homology equivalence, and generalized the results of homotopy regular morphism.In Chapter 2, we discussed the applications of homotopy regular morphism. Homeomorphic classification of all topology spaces is a difficult problem, while the graphlike manifolds are special topology spaces, their homeomorphic classification have been investigated by many authors. Specially, the cases with contraction Kn were observed, where Kn denotes the 1-skeleton of a (n-l)-simplex. Let Gn be the set of all the graphlike manifolds with contraction Kn, Bn the set of all the homeomorphic classes of Gn, and Pn the set of all the characteristic polynomials of the associated matrices of the graphlike manifolds in Gn. Let | Bn | and | Pn | be the cardinal number of Bn and Pn respectively. Hu Nuchen and ChenShengmin[30] pointed out that | Bn | > | Pn |, and raised a question: was | Bn | = | Pn | right or wrong? Guo Tuoying and Chen Shengmin[32] pointed out |Bn| = | Pn | for n < 6, and | B7 | = | P7 | = 54. In Section 2 of this chapter, we derived that | P8 | = 235, | P9| = 1824. So the low bound of homeomorphic classes were 235 and 1824 for G8 and G9 respectively. We also gave the representatives of the 235 homeomorphic classes for G8. In Section 3, we showed the canonical form about admissible operation of the associated matrices of graphlike manifolds with the contraction of Kn. In the last section, we discussed the week homeomorphic classification of graphlike manifolds.