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In this paper we study constructions of high dimensional knot groups from classical knot groups. We study certain homomorphic images of classical knot groups. Specifically, let K be a classical knot group and w any element in K. We are interested in the quotient groups G obtained by centralizing w, i.e. G = K/[K, w], and ask whether G is itself a knot group. For certain K and w we show that G can be realized as the group of a knotted 3-sphere in 5-space, but G is not realizable by a 2-sphere in 4-space. By varying w, we also obtain quotients that are groups of knotted 2-spheres in 4-space, but they cannot be realized as the groups of classical knots. We have examples of quotients K/[K,w] that have nontrivial second homology. Hence these groups cannot be realized as knot groups of spheres in any dimension. However, we show that these groups are groups of knotted tori in S4. Introduction. In this paper we study certain constructions of high dimensional knot groups from classical knot groups. As described by R. Litherland [L] the groups of twist spun knots and 2-knots obtained by the method of rolling are obtained from classical knot groups by centralizing peripheral elements. In an attempt to generalize the process of centralizing peripheral elements to produce 2-knot groups, we will study the general process of centralizing any element in a classical knot group. Thus if K is a classical knot group and w is any element in K, we study the quotient group G = K/ [K, w] and ask whether G is itself a knot group. We will show that the only obstruction to G being a high dimensional knot group is if H2(G) f 0. We have examples of a group K and elements w such that H2(K/[K,w]) 0. For certain K and w we show that G can be realized as the group of a knotted 3-sphere in 5-space, but G is not realizable by a 2-sphere in 4-space. By varying w, we also obtain quotients K/ [K, w] that are groups of knotted 2-spheres in 4-space. However these quotients cannot be realized as the groups of classical knots. 1. Preliminaries. In this paper we work in the smooth category. Sn denotes the n-sphere. An n-dimensional knot is a smooth embedding of Sn into Sn+2 and we denote it by (Sn+2, Sn). We refer to (Sn+2, SJn), n > 3, as a high dimensional knot, (S4, S2) as a 2-knot and (S3, S') as a classical knot. If K is a group and a, b are elements in K then the commutator [a, b] = a-'b-'ab. If w is any element in K then [K, w] is the smallest normal subgroup of K containing {[k, w]; k c K}. Similarly if A and B are subgroups of K then [A, B] denotes the Received by the editors August 1, 1985 and, in revised form, December 20, 1985. This paper was presented at the Spring Topology Conference of Florida State University, March 1985. 1980 Mathematics Subject Classification. Primary 57M25, 57Q45. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Construction of high-dimensional knot groups from classical knot groups