Torsion gaps in the homotopy of finite complexes (Q1113133)

A torsion gap of a space S is an interval [k,\(\ell]\) such that \(\Pi_ k(S)\otimes {\mathbb{Q}}\) and \(\Pi_{\ell}(S)\otimes {\mathbb{Q}}\) are non zero and each \(\Pi_ i(S)\otimes {\mathbb{Q}}=0\) if \(k<i<\ell\). If S is a finite CW-complex all known examples [k,\(\ell]\) of torsion gaps satisfy \(\ell - k<\sup \{i| \quad H_ i(S;{\mathbb{Q}})\neq 0\}=n(S).\) If \(\ell -k\geq n(S)\), call [k,\(\ell]\) a long torsion gap. In 1980, Y. Félix and H. Halperin proved that there are only finitely many long torsion gaps for a finite CW-complex. Here, the author gives limitation for the location and the length of long torsion gaps.