Smooth Homology Spheres and their Fundamental Groups

Let Mn be a smooth homology «-sphere, i.e. a smooth «-dimensional manifold such that H^(Mn)^H^(Sn).The fundamental group n of M satisfies the following three conditions:(1) 77 has a finite presentation,where Hfa) denotes the ith homology group of 77 with coefficients in the trivial Z7r-module Z. Properties (1) and ( 2) are trivial and (3) follows from the theorem of Hopf [2] which asserts that H2(tt) = H2(M)jpn2(M), where p denotes the Hurewicz homomorphism.For « > 4 we will prove the following converse Theorem 1.Let -n be a group satisfying the conditions (1), ( 2) and (3) above, and let n be an integer greater than 4.Then, there exists a smooth manifold Mn such that H*(Mn)^H*(Sn) and7TX(M)^tr.The proof is very similar to the proof used for the characterization of higher knot groups in [5].Compare also the characterization by K. Varadarajan of those groups 77 for which Moore spaces M(n, 1) exist [9].Not much seems to be known for « ^ 4. If M3 is a 3-dimensional smooth manifold with H*(M)^Hjf(S3), then 77=77^^^) possesses a presentation with an equal number of generators and relators.(Take a Morse function f on M with a single minimum and a single maximum.Then / possesses an equal number of critical points of index 1 and 2.) Also, under restriction to finite groups there is the following Theorem 2. Let M3 be a 3-dimensional manifold such that H*(M)^H*(S3).Suppose that ttx(M) is finite.Then, either tt1(M) = {1} or else, ttx(M) is isomorphic to the binary icosahedral group with presentation (x,y;x2 = f = (xv)5).This is implicitly well known: The hypotheses imply that 77=771(M) is a group of fix-point free transformations of a homotopy 3-sphere.Any such group belongs to a list established by Suzuki [8] and even to the shorter list of Milnor [7].The