3-Manifold invariants derived from the intersecting kernels

The intersecting kernel of a Heegaard splitting H1 boolean OR(S) H-2 for a compact orientable 3-manifold M is the subgroup K = Ker(i(1*)) boolean AND Ker(i(2*)) of pi(1)(S), where i(j*) : pi(1)(S) -> pi(1)(H-j) is the homomorphism induced by the inclusion i(j) : S curved right arrow H-j,H- j = 1, 2. In the paper, we obtain some invariants of 3-manifolds M from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.