Self maps of the suspension of H-spaces

The summary of research results is as follows.(A)Let X be an Hopf space。Oshima showed that the homotopy set of self maps of X[X，X]forms a group when X has at most 3-cells and he also determined the group structure of[X，X]for such spaces X.(B)We studied the group structure[ΣX，ΣX]，whereΣX is the suspension space of X.We determined the group structure in case that X=SU(3)and Sp(2)。They are typical example of above space X.In general，the suspension map[X，X]→[ΣX，∑X]is not a homorphism.(C)for general Hopf spaces，we can consider the Samelson products in[Y，X]when Y is a CW-complexes。The we found a formula which relates the Samelson product in[Y，X]and its suspension in[Y，ΩΣX]by using the generalized Hopf construction。The above item is an application of this formula.(D)It is important to know what nilpotency the group[X，X]for a Hopf space X.However another important is to study the composition structure of the group[X，X]。We determined this composition structure for X=SU(3)and Sp(2)。This structure looks like“Square ring”which Baues in Germany studied.(E)Let M I D1n文件D1=S I D1n-1文件D1U文件D22文件D2e文件D1n文件D1be the mod 2 Moore space。There is a canonical projection map P：M I D 1 n I D 1→S I D 1 n I D 1.Given an elementα∈πI D2κD 2(S I D 1 N I D 1)，we studied whenαhas a lift to M I D 1 n I D 1.One of our results is that the Whitehead square[li D2n i D2，lift for n≠1，3，or7.This is the joint work with J.Mukai in Shinshu University.

The summary of research results is as follows.(a)Let X be an Hopf space. Oshima showed that the homotopy set of self maps of X [X, X] forms a group when X has at most 3-cells and he also determined the group structure of [X, X] for such spaces X.(b)We studied the group structure [ΣX, ΣX], where ΣX is the suspension space of X. We determined the group structure in case that X = SU(3) and Sp(2). They are typical example of above space X. In general, the suspension map [X, X] → [ΣX, ΣX] is not a homomorphism.(c)For general Hopf spaces, we can consider the Samelson products in [Y, X] when Y is a CW-complexes. The we found a formula which relates the Samelson product in [Y, X] and its suspension in [Y, ΩΣX] by using the generalized Hopf construction. The above item is an application of this formula.(d)It is important to know what nilpotency the group [X, X] for a Hopf space X. However another important is to study the composition structure of the group [X, X]. We determined this composition structure for X = SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.(e)Let MィイD1nィエD1 = SィイD1n-1ィエD1 UィイD22ィエD2eィイD1nィエD1 be the mod 2 Moore space. There is a canonical projection map P : MィイD1nィエD1 → SィイD1nィエD1. Given an element α∈ πィイD2κィエD2(SィイD1nィエD1), we studied when α has a lift to MィイD1nィエD1. One of our results is that the Whitehead square [ιィイD2nィエD2, ιィイD2nィエD2] does not have a lift for n ≠1, 3, or 7. This is the joint work with J.Mukai in Shinshu University.