Celebratio Mathematica

C. M. Gordon: “Embedding piecewise linear manifolds with boundary,” Proc. Cambridge Philos. Soc. 72 : 1 (July 1972), pp. 21–25. MR 295359 Zbl 0236.57009 article

C. M. Gordon: “Knots in the 4-sphere,” Comment. Math. Helv. 51 (December 1976), pp. 585–596. MR 440561 Zbl 0346.55004 article

For convenience we work in the PL category. A knot of \( S^{n-1} \) in \( S^{n+1} \) is a locally flat submanifold of \( S^{n+1} \) homeomorphic to \( S^{n-1} \). The closure of the complement of a regular neighbourhood of a knot is its exterior. Two knots are equivalent if there is a homeomorphism of \( S^{n+1} \) taking one to the other. Equivalent knots therefore have homeomorphic exteriors. As far as the converse is concerned, it is known that for \( n\geq 3 \) there are at most two inequivalent knots with a given exterior [Gluck 1962; Browder 1967; Kato 1969; Lashof and Shaneson 1969; Swarup 1971]. Recently, Cappell and Shaneson have shown that for \( n = 4, 5 \), inequivalent knots with homeomorphic exteriors do exist [1976]. Their examples are certain knots whose exteriors fibre over \( S^1 \) with fibre \( T^n \)-open disc, where \( T^n \) is the \( n \)-dimensional torus (compare [Cappell 1970]). Since this approach uses the generalized Poincaré conjecture, however, in the case \( n = 3 \) it only yields knots in homotopy 4-spheres.

In the present paper, we use twist-spun knots to prove

There exist inequivalent knots \( K_1 \), \( K_2,\dots \), \( K_1^* \), \( K_2^*,\dots \) of \( S^2 \) in \( S^4 \), such that \( K_i \) and \( K_i^* \) have homeomorphic exteriors (\( i = 1 \), \( 2,\dots \)).

In the course of the proof, we show that removing a regular neighbourhood of a twist-spun knot in \( S^4 \) and sewing it back differently always gives \( S^4 \). In particular, this answers a question of Zeeman [1965, p. 494, problem 1], and enables us to give some new counterexamples to the 4-dimensional Smith conjecture [Giffen 1966; Gordon 1974; Sumners 1975].

A. J. Casson and C. M. Gordon: “On slice knots in dimension three,” pp. 39–53 in Algebraic and geometric topology (Stanford, CA, 2–21 August 1976), Part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics 32. American Mathematical Society (Providence, RI), 1978. MR 520521 Zbl 0394.57008 incollection

Under the equivalence relation of concordance (sometimes called cobordism), smooth knots in the 3-sphere \( S^3 \) form an abelian group with respect to connected sum [Fox and Milnor 1966]. The knots \( K \) representing the zero class are precisely those which are slice, that is, satisfy \[ (S^3,K) = \partial(B^4,D) \] for some smooth 2-disc \( D \) in the 4-ball \( B^4 \), Now associated with any knot \( K \) and a Seifert surface \( V \) spanning \( K \), is a bilinear Seifert pairing \[ \theta_V: H_1(V)\times H_1(V)\to \mathbf{Z} \] [Seifert 1935; Levine 1969]. We say that \( K \) is algebraically slice if \( \theta_V \) vanishes on a subgroup of \( H_1(V) \) whose rank is \( \frac{1}{2} \)rank \( H_1(V) \) (this condition is independent of the choice of \( V \)). It is known that a necessary condition for \( K \) to be slice is that it be algebraically slice. Moreover, in higher (odd) dimensions analogous definitions may be made, and there the conditions are equivalent [Levine 1969]. We shall show that this is not the case in dimension 3.

Andrew J. Casson	Related
Richard James Milgram	Related

A. J. Casson and C. M. Gordon: “A loop theorem for duality spaces and fibred ribbon knots,” Invent. Math. 74 : 1 (February 1983), pp. 119–137. MR 722728 Zbl 0538.57003 article

C. M. Gordon and R. A. Litherland: “Incompressible surfaces in branched coverings,” Chapter 7, pp. 139–152 in The Smith conjecture (New York, 6–7 April 1979). Edited by J. W. Morgan and H. Bass. Pure and Applied Mathematics 112. Academic Press (Orlando, FL), 1984. MR 758466 Zbl 0599.57006 incollection

Our primary aim is to show that if \( \widetilde{\Sigma} \) is the \( n \)-fold cyclic branched covering, \( n > 1 \), of a prime knot \( K \) in a homotopy 3-sphere \( \Sigma \), such that \( \Sigma - K \) is irreducible and contains a nonperipheral incompressible surface, then \( \widetilde{\Sigma} \) is not a homotopy sphere. This will be achieved by using the equivariant loop theorem to show that \( \widetilde{\Sigma} \) contains an incompressible surface of positive genus. Actually, we shall work in the more general context of a regular branched covering of a link in an arbitrary closed, orientable 3-manifold, as this is needed for the study of noncyclic finite group actions on homotopy 3-spheres. Our main result is Theorem 1, which asserts that, under certain circumstances, an incompressible surface in the complement of a link gives rise to one in any regular branched covering of the link. The special case of incompressible tori is considered in Theorem 2; here the proof actually uses the Smith conjecture. Interpreting Theorem 2 in terms of hyperbolic structures, using Thurston’s uniformization theorem for Haken manifolds (see Thurston [1979] and Chapter V, this volume by Morgan) we obtain Corollary 2.1, which states that if a regular branched covering of a link is hyperbolic, then so is the complement of the link.

These results are stated in Section 2, which also contains the necessary definitions and terminology. The proofs of Theorems 1 and 2 and Corollary 2.1 are given in Section 3. Finally, in Section 4, we give an elementary proof of the equivariant loop theorem for involutions (Theorem 3). A similar proof has been given by Kim and Tollefson [1980].

John Willard Morgan	Related
Richard Andrew Litherland	Related
Hyman Bass	Related

C. M. Gordon: “On the \( G \)-signature theorem in dimension four,” pp. 159–180 in À la recherche de la topologie perdue [In search of the lost topology]. Edited by L. Guillou and A. Marin. Progress in Mathematics 62. Birkhäuser (Boston), 1986. MR 900251 incollection

Alexis Marin	Related
Lucien Guillou	Related

A. J. Casson and C. M. Gordon: “Cobordism of classical knots,” pp. 181–199 in À la recherche de la topologie perdue [In search of the lost topology]. Edited by L. Guillou and A. Marin. Progress in Mathematics 62. Birkhäuser (Boston), 1986. With an appendix by P. M. Gilmer. MR 900252 incollection

Andrew J. Casson	Related
Alexis Marin	Related
Lucien Guillou	Related
Patrick M. Gilmer	Related

M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Dehn surgery on knots,” Ann. of Math. (2) 125 : 2 (1987), pp. 237–300. MR 881270 Zbl 0633.57006 article

John Edwin Luecke	Related
Marc Edward Culler	Related

A. J. Casson and C. M. Gordon: “Reducing Heegaard splittings,” Topology Appl. 27 : 3 (December 1987), pp. 275–283. MR 918537 Zbl 0632.57010 article

C. M. Gordon and J. Luecke: “Knots are determined by their complements,” J. Am. Math. Soc. 2 : 2 (1989), pp. 371–415. A much shorter version of this article was published in Bull. Am. Math. Soc. 20:1 (1989). MR 965210 Zbl 0678.57005 article

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Cameron McAllan Gordon

Cameron Gordon: a brief biography

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