Homologically trivial self-maps (Q1922157)

Let \(X\) be a CW-complex of the form \(X= S^n \cup e^{n+2} \cup e^{n+4}\) where \(n\geq 3\). In this paper the author computes the set of homotopy classes of self maps \(f:X \to X\) such that \(f\) induces the trivial map in singular homology. In order to do this, the author computes the homotopy groups \(\pi_{n+3} (A)\), \(\pi_{n+4} (A)\) of the skeleton \(A= S^n \cup e^{n+2}\). There are many different cases to consider, especially in the cases \(n=3\), \(n=4\), \(n=5\) and \(n\geq 6\) the behaviour is entirely different. The results are too technical to be stated here.