Co-h-space And Self-study, With The The Lun Equivalent Group

co-H-spaces are important objects of study of algebraic topology. The study ofself-homotopy equivalences is quite active in homotopy theory in the last twentyyears, co-H-spaces are dual of H-spaces, the latter had been a significant andcentral research contents in algebraic topology for many years in 20 century. A lot ofresults have been obtained and a sequence of theories have been set in this area andwhich become a important part of homotopy theory, co-H-spaces are popularspaces,which play an important role in algebraic topology, for example the spheresand suspension spaces are classical co-H-spaces. The authors P.J.Hilton,I.Berstein and G. Misilin ect. have investigated co-H-spaces in early years andhave gotten significant results. But the research of co-H-spaces is not deeply andwidely such as H-spaces. Now the situation is quite different in recent decade,M.Arkowitz and other authors take a systematic investigation in this field, includingrational co-H-spaces, the groups of self-homotopy equivalences ofco-H-spaces and other contents. D.W.kahn has pointed out 17 research problemsof self-homotopy equivalences in 1988, the 12th. problem is on the self-homotopyequivalences of co-H-spaces. Recently the study of the groups of self-homotopyequivalences of co-H-spaces is not sufficient. In this thesis we studyco-H-spaces in many fields, including the homology decomposition of spaces,rational co-H-spaces and related topics, the special properties of co-H-mapsof co-H-spaces, the structure of the homotopy set [X, Y] of co-H-spaces, theself-homotopy equivalences of co-H-spaces, the factorizations of subgroups ofself-homotopy equivalences and so on. We also approach the group of self-homotopyequivalences of some spaces and obtain some results. The homology decomposition isdual to homotopy decomposition, i.e. Posnikov system. Posnikov system is widelyused in homotopy theory, and is one of important tools in the study of homotopytheory. It is known that the homology decomposition is not complete such ashomotopy decomposition is, but it is complete in some spaces and categories, forexample, the space of wedge of spheres and rational co-H-spaces. We make someresearch work in it and have some results. F-equivalence and F-isomophisim havebeen discussed in this thesis. When X is co-H-space, for any space Y thehomotopy set [X, Y] has a binary operation with the co-H-structure on X andthe homotopy set [X,Y] can become a group when this co-H-structure is aco-H-group structure. We study some properties of co-H-maps ofco-H-spaces and the structure of some subgroups of [X, Y], we have obtained asequence of results in this field. The group of self-homotopy equivalences havebeen studied in this thesis also, including the structures of the subgroup Aut_co-H(SX) and Aut_S(SX)of Aut(SX)or subgroup Aut_co-H(X) of Aut(X) about co-H-spaces X, and the subgroup Aut_*(X), Aut_#N(X), Aut_∑(X), Aut_Ω(X) of Aut(X)about CW-spaces X. Some results have been gotten in this field.There are five chapters in this thesis. Chapter one is introduction. Chapter two isabout definitions, F-equivalence and F-isomophism are discussed in this chapter.Chapter three is the investigation of homology decomposition, some results have beenshown. In chapter four we study the special properties of co-H-maps ofco-H-spaces and the structure of some subgroups of [X, Y], we have obtained asequence of important results in this chapter. We approach the group of self-homotopy equivalences in chapter five and some results have been obtained.