M-fibrewise homotopy extension property and its applicationos

Some homotopical notions in M-fibrewise category such as M-fibrewise homotopy extension property are studied. With the help of notions such as M-fibreswise retract and M-fibrewise deformation retract, an equivalent description of M-fibrewise homotopy extension property which generalizes the related one for homotopy extension property in general topological category is given. Furthermore the homotopy invariance of some constructions in M-fibrewise category such as M-fibrewise attaching space is shown. Finally, a judgment theorem is given via M-fibrewise mapping cylinder for the problem on whether two M-fibrewise maps are homotopic. This judgment is a natural generalization for the one on whether two maps are homotopic in general topological category.