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Equivariant concordance of invariant knots


Author: Neal W. Stoltzfus
Journal: Trans. Amer. Math. Soc. 254 (1979), 1-45
MSC: Primary 57Q45
DOI: https://doi.org/10.1090/S0002-9947-1979-0539906-5
MathSciNet review: 539906
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Abstract: The classification of equivariant concordance classes of high-dimensional codimension two knots invariant under a cyclic action, T, of order m has previously been reported on by Cappell and Shaneson [CS2]. They give an algebraic solution in terms of their algebraic k-theoretic $ \Gamma $-groups. This work gives an alternative description by generalizing the well-known Seifert linking forms of knot theory to the equivariant case. This allows explicit algorithmic computations by means of the procedures and invariants of algebraic number theory (see the subsequent work [St], particularly Theorem 6.13). Following Levine [L3], we define bilinear forms on the middle-dimensional homology of an equivariant Seifert surface $ {B_i}(x,y) = L(x,{i_ + }(T_{\ast}^iy))$, for $ i = 1, \cdots ,m$. Our first result (2.5) is that an invariant knot is equivariantly concordant to an invariant trivial knot if and only if there is a subspace of half the rank on which the $ {B_i}$ vanish simultaneously. We then introduce the concepts of equivariant isometric structure and algebraic concordance which mirror the preceding geometric ideas. The resulting equivalence classes form a group under direct sum which has infinitely many elements of each of the possible orders (two, four and infinite), at least for odd periods. The central computation (3.4) gives an isomorphism of the equivariant concordance group with the subgroup of the algebraic knot concordance group whose Alexander polynomial, $ \Delta $, satisfies the classical relation $ \left\vert {\prod\nolimits_{i = \,1}^m {\Delta \left( {{\lambda ^i}} \right)} } \right\vert\, = \,1\,$, where $ \lambda $ is a primitive mth root of unity. This condition assures that the m-fold cover of the knot complement is also a homology circle, permitting the geometric realization of each equivariant isometric structure. Finally, we make an explicit computation of the Browder-Livesay desuspension invariant for knots invariant under an involution and also elucidate the connection of our methods with the results of [CS2] by explicitly describing a homomorphism from the group of equivariant isometric structures to the appropriate $ \Gamma $-group.


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  • [B1] H. Bass, The Dirichlet unit theorem, induced characters and Whitehead groups of finite groups, Topology 4 (1965), 391-410. MR 0193120 (33:1341)
  • [B2] -, $ {K_2}$ des corps globaux, Seminaire Bourbaki, 23e annee, 1970/71, no. 394.
  • [Bla] R. C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340-356. MR 0085512 (19:53a)
  • [Br1] W. Browder, Manifolds with $ {\pi _{1\,}}\, = \,Z$, Bull. Amer. Math. Soc. 72 (1966), 234-244. MR 0190940 (32:8350)
  • [Br2] -, Surgery and the theory of differentiable transformation groups, Proc. Tulane Symposium on Transformation Groups, Springer, Berlin, 1968, pp. 1-46. MR 0261629 (41:6242)
  • [Br3] -, Free $ {Z_p}$-actions on homotopy spheres, Topology of Manifolds, Markham, Chicago, Ill., 1970, pp. 217-226.
  • [Br4] -, Surgery on simply-connected manifolds, Springer, New York, 1972. MR 0358813 (50:11272)
  • [BL] W. Browder and G. R. Livesay, Fixed point free involutions on homotopy spheres, Tôhoku Math. J. 25 (1973), 69-88. MR 0321077 (47:9610)
  • [BPW] W. Browder, T. Petrie and C. T. C. Wall, The classification of free actions of cyclic groups of odd order on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 455-459. MR 0279826 (43:5547)
  • [Bu] E. Burger, Über gruppen mit verschlingungen, J. Reine Angew. Math. 188 (1950), 193-200. MR 0043089 (13:204a)
  • [CS1] S. Cappell and J. Shaneson, Submanifolds, group actions and knots. I, II, Bull. Amer. Math. Soc. 78 (1972), 1045-1052. MR 0383432 (52:4313)
  • [CS2] -, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277-348. MR 0339216 (49:3978)
  • [D] A. Dold, Lectures on algebraic topology, Springer, Berlin, 1972. MR 0415602 (54:3685)
  • [FM] R. H. Fox and J. Milnor, Singularities of two spheres in four space and cobordism of knots, Osaka J. Math. 3 (1966), 257-267. MR 0211392 (35:2273)
  • [F1] R. H. Fox, Free differential calculus. III: Subgroups, Ann. of Math. (2) 64 (1956), 407-419. MR 0095876 (20:2374)
  • [F2] -, On knots which are fixed under a periodic transformation of the 3-sphere, Osaka J. Math. 10 (1958), 31-35. MR 0131872 (24:A1719)
  • [F3] -, A quick trip through knot theory, Topology of Three Manifolds and Related Topics (M. K. Fort, Jr., editor), Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 0140099 (25:3522)
  • [Fu] F. B. Fuller, A relation between degree and linking numbers, Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, N. J., 1957, pp. 258-262. MR 0084137 (18:815c)
  • [G] C. McA. Gordon, Knots whose branched cyclic coverings have periodic homology, Trans. Amer. Math. Soc. 168 (1972), 357-370. MR 0295327 (45:4394)
  • [H] A. Hatcher, Concordance and isotopy of smooth embeddings in low codimensions, Invent. Math. 21 (1973), 223-232. MR 0334251 (48:12570)
  • [Hi] F. E. P. Hirzebruch, Singularities and exotic spheres, Seminaire Bourbaki, 19e annee, 1966/67, no. 314.
  • [KaMa] M. Kato and Y. Matsumoto, Simply connected surgery of submanifolds in codimension two, J. Math. Soc. Japan 24 (1972), 586-608. MR 0307249 (46:6369)
  • [K1] M. Kervaire, Les noeuds de dimension superieure, Bull. Soc. Math. France 93 (1965), 225-271. MR 0189052 (32:6479)
  • [K2] -, Knot cobordism in codimension two, Lecture Notes in Math., vol. 197, Springer, Berlin, 1971, pp. 83-105. MR 0283786 (44:1016)
  • [KM] M. Kervaire and J. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504-537. MR 0148075 (26:5584)
  • [KV] M. Kervaire and A. Vasquez, Simple connectivity and the Browder-Novikov theorem, Trans. Amer. Math. Soc. 126 (1967), 508-513. MR 0208602 (34:8411)
  • [La] S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965. MR 0197234 (33:5416)
  • [Lef] S. Lefschetz, Algebraic topology, Amer. Math. Soc. Colloq. Publ., vol. 27, Amer. Math. Soc., Providence, R. I., 1942. MR 0007093 (4:84f)
  • [L1] J. Levine, Unknotting spheres in codimension two, Topology 4 (1965), 9-16. MR 0179803 (31:4045)
  • [L2] -, Polynomial invariants of knots in codimesion two, Ann. of Math. (2) 84 (1966), 537-554. MR 0200922 (34:808)
  • [L3] -, Knot cobordism in codimension two, Comment. Math. Helv. 44 (1968), 229-244.
  • [L4] -, Invariants of knot cobordism, Invent. Math. 8 (1969), 98-110. MR 0253348 (40:6563)
  • [L5] -, An algebraic classification of some knots of codimension two, Comment. Math. Helv. 45 (1970), 185-198. MR 0266226 (42:1133)
  • [LdM1] S. Lopez de Medrano, Involutions on manifolds, Springer, New York, 1971. MR 0298698 (45:7747)
  • [LdM2] -, Nudos invariantes bajo involuciones. I, An. Inst. Mat. Univ. Nac Autónoma de México 8 (1969), 81-90.
  • [LdM3] -, Invariant knots and surgery in codimension two, Actes Congres Internat. Math. 2, Gauthier-VIllars, Paris, 1970.
  • [Ma] Y. Matsumoto, Knot cobordism groups and surgery in codimension two, J. Fac Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 253-317. MR 0334228 (48:12547)
  • [M1] J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137-147. MR 0141115 (25:4526)
  • [M2] -, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 0196736 (33:4922)
  • [M3] -, Infinite cyclic coverings, Conference on the Topology of Manifolds (J. G. Hocking, editor), Prindle, Weber and Schmidt, Boston, Mass. 1968, pp. 115-133. MR 0233363 (38:1685)
  • [M4] -, On isometries of inner product spaces, Invent. Math. 8 (1969), 83-97. MR 0249519 (40:2764)
  • [M5] -, Introduction to algebraic K-theory, Ann. of Math. Studies, No. 72, Princeton Univ. Press, Princeton, N. J., 1971. MR 0349811 (50:2304)
  • [M6] -, Lectures on the h-cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton Univ. Press, Princeton, N. J., 1965. MR 0190942 (32:8352)
  • [Rei] K. Reidemeister, Durschnitt und schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226-239. MR 0000634 (1:105h)
  • [Ro] R. A. Robertello, An invariant of knot cobordism, Comm. Pure Appl. Math. 18 (1965), 543-555. MR 0182965 (32:447)
  • [Sch] R. E. Schultz, Smooth structures on $ {S^{p\,}}\, \times \,{S^q}$, Ann. of Math. (2) 90 (1969), 187-198. MR 0250321 (40:3560)
  • [Sei] H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1934), 571-592. MR 1512955
  • [St] N. Stoltzfus, Unraveling the integral knot concordance group, Mem. Amer. Math. Soc., No. 192 (1977). MR 0467764 (57:7616)
  • [Sh1] J. Shaneson, Embeddings with codimension two of spheres in spheres and h-cobordisms of $ {S^{1\,}}\, \times \,{S^3}$, Bull. Amer. Math. Soc. 74 (1968), 972-974. MR 0230325 (37:5887)
  • [Sh2] -, Wall's surgery obstruction group for $ Z\, \times \,G$, Ann. of Math. (2) 90 (1969), 296-334. MR 0246310 (39:7614)
  • [Sp] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [T] H. F. Trotter, On S-equivalence of Seifert matrices, Invent. Math. 20 (1973), 173-207. MR 0645546 (58:31100)
  • [W1] C. T. C. Wall, Diffeomorphisms of four-manifolds, Proc London Math. Soc. 39 (1964), 131-140. MR 0163323 (29:626)
  • [W2] -, Surgery on compact manifolds, Academic Press, New York, 1970. MR 0431216 (55:4217)
  • [W3] -, Quadratic forms on finite groups. II, Bull. London Math. Soc. 4 (1972), 156-160. MR 0322071 (48:435)

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DOI: https://doi.org/10.1090/S0002-9947-1979-0539906-5
Keywords: Knots, linking numbers, concordance, cyclic action, codimension two embeddings, lens spaces, isometric structures
Article copyright: © Copyright 1979 American Mathematical Society

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