Cancellation and Non Cancellation Phenomena for Infinite Complexes

Abstract

Let $L(I,J)$ be defined by the pull-back of $C{P}_1^{\infty }\to K(Q,2)\leftarrow \Omega S_J^3$ where $\{I,J\}$ is a partition of all primes. We classify spaces $\{{\Omega }^k{\Sigma }^{k}C(f)\}$ of loop-suspension of mapping cone of phantom map $f:L(I,J)\to S^3$ for $k=0,1,\cdots ,\infty$ which have the same $n$-type for all $n$. In the category of finite CW-complexes, cancellation and non cancellation phenomena are well studied. In the category of infinite CW-complexes, the phenomena are less known. We study cancellation and non cancellation phenomena for infinite dimensional complexes by using above spaces.