Invariants of 3 - Manifolds

The ju-invariant fi(M) of an oriented Z2 -homology 3-sphere M is defined by Hirzebruch in [8], using Rohlin's Theorem [13], to be the mod 16 reduction of the signature of a framed manifold W with M = dW. In this paper we give a formula for p(M) by studying M as a branched dihedral covering space of S 3 . Hilden [7] and Montesinos [9] have independently shown that every closed orientable 3-manifold is actually a 3-fold (dihedral) covering space of S branched along a knot. Also see [1], [6] and [12]. Let a be a smooth or piecewise linear oriented knot S C S. Let V C S with dV = a be a Seifert surface for a. The Seifert form L = Lv is the bilinear form of Unking numbers of circles in V, with respect to a fixed orientation of S 3 , and L' is given as L'(x, y) = L(y, x). Let p be an odd integer. A knot j3 will be called a modp characteristic knot (in V) of a if there is an embedding of S 1 in V, with nontrivial homology class [|8] E Ht(V), so that the composite S 1 C V C S is ft and if L(x, 0) + L(p, x) = 0 (modp) fo ra l lx inT/^K) . A modp characteristic knot 0 for a determines a homomorphism p of n^S a) onto the dihedral group Z 2 x w Zp of order 2p. The map p is characterized by the requirements that its composition with Z2 x w Zp —• Z 2 be nontrivial and that, for x in the image of TIX V9 p(x) G Zp C Z2 x w Zp is the modp reduction of L(x, )3). Hence ]3 determines a p-fold dihedral branched covering Ma^ of 5 3 , branched along a. It can be shown that every dihedral representation for a and associated branched cover of S are determined by a characteristic knot for a in V. Further, dihedral representations can easily be classified in terms of equivalence classes of characteristic knots. By abuse of notation, we write Ma for Ma^\ as "most" knots have at most one (up to conjugacy) dihedral representation of order 2p, this notation is usually strictly justified.