Some Problems On Homotopy Theory

Topological space and homotopy classification of continuous mapping are important contents in the algebraic topology. At first, we study the homeomorphic class of graphlike manifold with contraction round diagrams and its class number. Rround diagram takes a vertex as centre, other n-1vertexes link together one after another to form a circle. Each vertexes are linked with the centre. We obtain the number of homeomorphic class of Wn*by means of classification, use the software Matlab to compute the adjoint matrix of Wn*and calculate the number of its characteristic polynomial, and then get the lower bound of the number of homeomorphic class of Wn*. We also put forward a method to calculate the upper bound of the number of homeomorphic class of Wn*by using Burnside lemma. We take the round diagram W13for example, and then we use above methods to calculate the homeomorphic class number of all graphlike manifold W13*with contraction round diagrams W13is224, its upper bound of homeomorphic class number is224, and its lower bound of homeomorphic class number is214. The homeomorphic class number of all graphlike manifold W13*with contraction round diagrams W13and its upper bound of homeomorphic class number is same. Then we calculate the homeomorphic class number of all graphlike manifold Wn*(n=5,…,12,14) with contraction round diagrams Wn(n=5,…,12,14) and its upper bound of homeomorphic class number, find that they are also equal. So we put forward a conjecture:if n≥5, the homeomorphic class number of graphlike manifold Wn*with contraction round diagrams Wn is as same as its upper bound which obtained by using Burnside lemma.At the second part, we mainly study on homotopy regular morphism in the category of topological pairs. We generalize homotopy monomorphism, homotopy epimorphism and homotopy regular morphism in the category of topological spaces(TOP*) to the category of topological pairs (TOPA). The article provides the conditions which make homotopy regular morphism exit in the category of topological pairs, gets the relationship between homotopy regular morphism and homotopy monomorphism, homotopy epimorphism, left or right invertible and homotopy equivalence in the category of topological pairs, and points out a kind of property of homotopy regular morphism in the category of topological pairs. We generalize the concepts homotopy monomorphism, homotopy epimorphism and homotopy regular morphism in the category of topological pairs to stable homotopy monomorphism, stable homotopy epimorphism and stable homotopy regular morphism. And then study the conditions which make stable homotopy regular morphism exit and the properties, discuss the relationship between stable homotopy regular morphism and stable homotopy monomorphism, stable homotopy epimorphism and stable homotopy equivalence in the category of topological pairs.