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