Journal of the
Korean Mathematical Society JKMS

In this paper, we extend the concept of the group $ \mathcal{E} (X)$ of self homotopy equivalences of a space $X$ to that of an object in the category of pairs. Mainly, we study the group $\mathcal{E}(X, A)$ of pair homotopy equivalences from a CW-pair $(X,A)$ to itself which is the special case of the extended concept. For a CW-pair $(X,A)$, we find an exact sequence $1\rightarrow G \rightarrow \mathcal{E}(X, A)\rightarrow \mathcal{E}(A)$ where $G$ is a subgroup of $\mathcal{E}(X, A)$. Especially, for CW homotopy associative and inversive $H$-spaces $X$ and $Y$, we obtain a split short exact sequence $1\rightarrow \mathcal{E}(X) \rightarrow \mathcal{E}(X \times Y, Y)\rightarrow \mathcal{E}(Y)\rightarrow 1$ provided the two sets $[X\wedge Y,X\times Y]$ and $[X, Y]$ are trivial.

Keywords: self homotopy equivalence, self pair homotopy equivalence

MSC numbers: Primary 55P10; Secondary 55P30, 55P20