PRINCIPAL COFIBRATIONS AND GENERALIZED CO-H-SPACES

For a map $p:X{\rightarrow}A$, there are concepts of co-$H^p$-spaces, co-$T^p$-spaces, which are generalized ones of co-H-spaces [17,18]. For a principal cofibration $i_r:X{\rightarrow}C_r$ induced by $r:X^{\prime}{\rightarrow}X$ from $\imath:X^{\prime}{\rightarrow}cX^{\prime}$, we obtain some sufficient conditions to having extensions co-$H^{\bar{p}}$-structures and co-$T^{\bar{p}}$-structures on $C_r$ of co-$H^p$-spaces and co-$T^p$-structures on X respectively. We can also obtain some results about co-$H^p$-spaces and co-$T^p$-spaces in homology decompositions for spaces, which are generalizations of Golasinski and Klein's result about co-H-spaces.