@article{Shore1979TheHC,
title={The homogeneity conjecture.},
author={Richard A. Shore},
journal={Proceedings of the National Academy of Sciences of the United States of America},
year={1979},
volume={76 9},
pages={4218-9}
}

We show that, for any function f in which Kleene's O is computable, the ordering of Turning degrees (i.e., degrees of difficulty of computation of functions) is not isomorphic to the ordering of degrees of functions from which f is computable. This refutes a well-known conjecture of H. Rogers, Jr., and others.