On the Classification of $(n-2)$-connected $2n$-manifolds with Torsion Free Homology Groups

C.T.C. Wall investigated the classification problems of highly connected manifolds in his series of papers Q18], Q19], and pi]. But since then only a few investigations to classify the manifolds with lower connectedness have been made. As such investigations there are Wall [J20], Tamura Q17], and Ishimoto T6]. In this paper, we consider to classify (n — 2)-connected 2 Tz-manifolds which have torsion free homology groups (equivalently, torsion free (n — l)-th homology groups) and are (n — l)-parallelizable. Such a manifold can be decomposed as a connected sum of an (n — 2)connected 2n,-manifold which has the vanishing n,-th homology group and is {n — l)-parallelizable and an (n — l)-connected manifold. So that, our main problems are to classify the former 2 n,-manifolds and to investigate the uniqueness of the decomposition. Firstly, we completely classify the handlebodies of 3?(2ra + l, k, ra + 1) (zi^4) up to diffeomorphism, and then, using the results we consider to classify (n — 2)-connected 2 n -manifolds which have vanishing n-th homology groups and are (n — l)-parallelizable up to diffeomorphism mod02w Here, 3?(m, k, s) is the collection of k handlebodies W=D \J { \J DfxDf'} with the disjoint smooth imbeddings {/i> «=i /,-: dD{ xDf~ -»dD, i = l, 2 , , k. The uniqueness of the decomposition is also considered up to diffeomorphism mod02w Throughout this paper, manifolds are connected, closed, and differentiate. I would like to express my hearty thanks to Professor N. Shimada for his kind encouragement and advices during the preparation of this paper.