For a simply connected CW-complex \(X\), the homology self-closeness number is the minimum non-negative integer \(k\) such that any self-map \(f\) of \(X\) inducing an isomorphism on homology groups up to the dimension \(k\) is a self-homotopy equivalence. This number is homotopy invariant and closely related to the group of self-homotopy equivalences.
We compare the homology self-closeness numbers between two simply connected CW-complexes that fit into a cofibration sequence. For a simply connected CW-complex \(X\), we recall the homology decomposition \(X_n\) and homotopy decomposition \(X^{(n)}\), respectively. We study the homology self-closeness numbers for \(X_n\) and \(X^{(n)}\) and establish several relations among the homology self-closeness numbers of \(X, X_n\) and \(X^{(n)}\). Finally, we apply these results to compute some examples.
