SELF-CLOSENESS NUMBERS OF NON-SIMPLY-CONNECTED SPACES

The self-closeness number of a connected CW complex is the least integer n such that any of its self-maps inducing an isomorphism in 7r,k for * <= n is a homotopy equivalence. We prove that under a mild condition, the self-closeness number of a non-simply-connected finite complex coincides with that of its universal cover whenever the universal cover is a finite H0-space or a finite co-H0-space. We give several interesting examples to which the result applies.