Equivariant concordance of invariant knots

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 Γ \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 ∗ i y ) ) {B_i}(x,y) = L(x,{i_ + }(T_{\ast }^iy)) , for i = 1 , ⋯ , m 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 {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, Δ \Delta , satisfies the classical relation | ∏ i = 1 m Δ ( λ i ) | = 1 \left | {\prod \nolimits _{i = \,1}^m {\Delta \left ( {{\lambda ^i}} \right )} } \right |\, = \,1\, , where λ \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 Γ \Gamma -group.