Recently M. Mather has
generalized results of T. Ganea concerning homotopy fibers and cofibers. In this
paper we present two results on cofibers, one of which substantially extends and
clarifies Mather’s generalization. Our other result (which is used to prove the
first) in part examines the mapping cone of the fiberwise join of two maps.
Applications of the results are made to reprove a result of I. M. James on the
fiberwise suspension and to give a characterization of coreducible Thom
spaces.