Let be a torus embedded in *S and X its exterior. The peripheral subgroup of is the image of nt(dX) by the map /„ induced by the inclusion i: dXc+X. So it is isomorphic to the direct sum of the infinite cyclic group Z, which is generated by a meridian, and some quotient of π1(F) = Z®Z. We denote by τF the second summand and call it the type of F see [10, §3]. If is unknotted, that is, bounds a solid torus in S 4 (see [7]), then τ = 0 . If is a torus constructed by spinning a non-trivial classical knot, then τF=Z, cf. [3, 10, 11]. Asano [1] and Litherland [10] constructed examples with τ = Z φ Z . And Boyle [3] showed that there are tori of type Zn for n = 2,5 and 10 by attaching a 2-dimensional 1-handle to the 5-twist-spun trefoil. We abbreviate the group π^S^ — F) of as πF. It is known that the second homology of the group H2(nF) is a quotient of H2(S ' — F) = Z φ Z, and several authors gave examples having non-trivial second homology [2, 4], see also [12]. Litherland [10] showed that H2(πF) is a quotient of τF and that any quotient of Z φ Z is realizable as H2{πF) for some torus of type Z φ Z . For abelian groups A and JB, we write A <B if A is a quotient of B. It is natural to ask