(Co)homology self-closeness numbers of simply-connected spaces

Abstract

The (co)homology self-closeness number of a simply-connected based CW-complex X is the minimal number k such that any self-map f of X inducing an automorphism of the (co)homology groups for dimensions ≤ k is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology self-closeness numbers of spaces in certain cofibrations, define the mod p (co)homology self-closeness number of simply-connected p-local spaces with finitely generated homologies and study some properties of the (mod p) (co)homology self-closeness numbers.