Homology of groups of surfaces in the 4-sphere

1. Let F be a closed, connected, orientable surface of genus g ≥ 0 smoothly embedded in S4, and let π denote the fundamental group π1(S4 − F). Then H2(π) is a quotient of H2(S4 − F) ≅ H1(F) ≅ Z2g. If F is unknotted, that is, if there is an ambient isotopy taking F to the standardly embedded surface of genus g in S3 ⊂ S4, then π ≅ Z, so H2(π) = 0. More generally, if F is the connected sum of an unknotted surface and some 2-sphere S, then π ≅ π1 (S4 − S), so again H2(π) = 0. The question of whether H2(π) could ever be non-zero was raised in (5), Problem 4.29, and (10), Conjecture 4.13, and answered in (7) and (1). There, surfaces are constructed with H2(π)≅ Z/2, and hence, by forming connected sums, with H2(π) ≅ (Z/2)n for any positive integer n. In fact, (1) produces tori T in S4 with H2(π) ≅ Z/2, and hence surfaces of genus g with H2(π) ≅ (Z/2)g.