Simply Connected Five-Manifolds

Abstract

S. Smale, using his theory of handlebodies, has classified, under diffeomorphism closed, simply connected, smooth 5-manifolds with vanishing second Stiefel-Whitney class. C.T.C. Wall has given a classification of (n 1)-connected (2n + 1)-manifolds which does not however cover the case n = 2. In this paper we complete the classification of simply connected 5-manifolds. A.A.Markov has proved that a general classification of 5-manifolds is impossible, but it seems reasonable to hope for results in the case of 5-manifolds with a given fundamental group. The second Stiefel-Whitney class of a simply connected manifold may be regarded as a homomorphism w: H2(M; Z) Z, and we may arrange w to be non-zero on at most one element of a 'basis' (0.5), this element having order 2i for some i (Lemma C). Then i is a diffeomorphism invariant i(M) of M. If H2(X) -H(M), and i(X) = i(M), where X and M are simply connected 5-manifolds, then there are (0.8) isomorphisms 0: H2(X) H2(M) which preserve the linking form b on the torsion subgroups (0.7), and which satisfy w(M)oO = w(X). The basic theorem (2.2) states that any such isomorphism may be realized by a diffeomorphism of X with M. Thus H2(M) and i(M) form a complete set of invariants for the diffeomorphism classification. On the other hand b imposes restrictions on the second homology group (Lemma E), and hence on the decomposability of the manifolds. Using results of C.T.C.Wall on diffeomorphisms of 4-manifolds, it is possible to construct an example of an indecomposable manifold for each possible homology group (? 1) and, using these, to give a canonical manifold in each diffeomorphism class (Theorem 2.3). In addition to the main theorems, ?2 contains some corollaries and applications of them. The manner of construction of the indecomposable manifolds and manifolds similar to them produces minimal handle decompositions and allows the computation of embedding and immersion dimensions. The nature of the invariants also allows an extension of the results. The proof of Theorem 2.2 is omitted from ? 2 and occupies the remainder of the paper. X and M as above are necessarily cobordant (Lemma F), and the first step is to find a cobordism with minimal homotopy groups (? 3), i.e., one which is simply connected and with second homology group zero or, if iw(X) # 0, Z2. In ?4 modifications are described of which one enlarges the