The Gromov-lawson-rosenberg Conjecture for Cocompact Fuchsian Groups

We prove the Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature. Given a smooth closed manifold M, it is a long-standing question to determine whether or notM admits a Riemannian metric of positive scalar curvature. Work of Gromov-Lawson and Schoen-Yau shows that if N admits positive scalar curvature and M is obtained from N by k-surgeries of codimension n−k ≥ 3, then M admits positive scalar curvature as well. In the case when M is spin, this surgery result implies the following. Bordism Theorem. ([10], [27]) Let M be a closed spin manifold, n ≥ 5, G = π1(M), and suppose u : M → BG induces the identity on the fundamental group. If there is a positively scalar curved spin manifold N and a map v : N → BG such that [M,u] = [N, v] ∈ Ω n (BG), then M admits a metric of positive scalar curvature. On the other hand, the work of Lichnerowicz gives an obstruction to manifolds admitting positive scalar curvature. Using the Weitzenböck formula for the Dirac operator and the Atiyah-Singer index theorem, he proves in [18] that if M is a closed spin manifold with positive scalar curvature then Â(M), the A-hat genus of M , vanishes. Generalizations of the Dirac operator and its index by Hitchin [12] and Mǐsčenko-Fomenko [20] provide obstructions as well, taking final form in the following theorem of Rosenberg [23]. Obstruction Theorem. [23] [29] Let M be a closed spin manifold and u : M → BG be a continuous map for some discrete group G. If M admits a metric of positive scalar curvature then α[M,u] = 0 in KOn(C ∗ rG), where α : Ω Spin n (BG) → KOn(C ∗ rG) is the index of the Dirac operator. Here C rG is the reduced real C -algebra of G, a suitable completion of the group algebra RG. In dimensions 4k with G = 1, the index α[M,u] agrees with Â(M) up to a constant factor ([17], p. 149). The above theorem motivates The Gromov-Lawson-Rosenberg (GLR) Conjecture for G. Let M be a closed spin manifold, n ≥ 5, G = π1(M), and suppose u : M → BG induces the 2000 Mathematics Subject Classification Primary: 53C21. Secondary: 19L41, 19L64, 57R15, 55N15, 53C20.