The Homotopy Type of a Poincaré Duality Complex After Looping

Abstract

Abstract We answer a weaker version of the classification problem for the homotopy types of ( n — 2)-connected closed orientable (2 n — 1)-manifolds. Let n ≥ 6 be an even integer and let X be an ( n — 2)-connected finite orientable Poincaré (2 n — 1)-complex such that H n-1 ( X ;ℚ) = 0 and H n-1 (X; ℤ 2 ) = 0. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on H n-1 ( X ; ℤ p ) for each odd prime p . A stronger result is obtained when localized at odd primes.