Equivariant Homotopy Epimorphisms And Monomorphisms

Homtopy theory is the study of topological spaces using a broader equivalence than homeomorphism. It's really the natural setting for algebraic topology because the traditional algebraic invariants (fundamental group, homolog group,etc.) arc isomorphic not only for homeomorphic spaces but for spaces of the same homotopy type. Homtopy epimorphisms and monomorphisms are special morphisms in the category of topological spaces and the initial research of them may be traceable to S.T.Hu's book [14] published in 1959. In that book he first proved that the Hopf fibration S~3→S~2 is a homotopy monomorphism and used it to solve the classification problem of maps from triangulated 3-spaces to S~2. In 1965, It is Hilton who first introduced the concepts of homotopy epimorphisms and monomorphisms in [16]. From then on, more and more algebraic topologist become interested in this area. On the other hand, equivariant algebraic topology concerns the study of algebraic invariants of space with group actions. It is really a study of symmetries on spaces, and has always occupied a central role in mathematics. The research of equivariant epimorphisms and monomorphisms is just the intersection of the above two interests.In this thesis, we first deal with the basic notions and elementary results in the theory of homotopy epimorphisms and monomorphisms. Then, we describe the basic definitions of the equivariant homotopy theory, with some important results for our future purpose. After that, the concepts of equivariant homtopy pullbacks and pushouts are initially introduced and we will use them to characterize equivariant homotopy epimorphisms and monomorphisms.Moreover, the effect of the equivariant homotopy pullback and pushout are investigated while their passing to their H-fixed point spaces. Then we discuss the equivariant fibration and several results about the ordinary fibration are generalized to the equivariant category. Thus we may use them to get an equivariant version of the cube theorem and establish the result that equivariant homotopy pullbacks preserve equivariant homotopy epimorphism, which generalize Shen's an important theorem [19] in 1994.In additon, we also get several results of equivariant homotopy epimorphism and monomorphism.