Unified So(3) Quantum Invariants for Rational Homology 3–spheres

Let M be a rational homology 3–sphere with |H1(M,Z)| = b. For any odd divisor c of b, we construct a unified invariant IM,c lying in a cyclotomic completion of a certain polynomial ring, which dominates Witten–Reshetikhin–Turaev SO(3) invariants of M at all roots of unity whose order r satisfies (r, b) = c. For c = 1, we recover the unified invariant constructed by Le and Beliakova–Le. If b = 1, our invariant coincides with Habiro’s invariant of integral homology 3–spheres. New structural properties of the set of quantum invariants at roots of unity not coprime to the torsion are the main applications of our construction. Introduction The SO(3) Witten–Reshetikhin–Turaev invariant τM (ξ) ∈ Q(ξ) is defined for any closed oriented 3–manifold M and any root of unity ξ of odd order [16], [8]. If, in addition, the order of ξ is prime, then by the results of Murakami [15] andMasbaum–Roberts [13], τM (ξ) is an algebraic integer. This integrality result was the starting point for the construction of integral TQFTs, representations of the mapping class group over Z[ξ] [5], and categorification of quantum 3–manifold invariants [9]. The proofs in [15] and [13] depend heavily on the arithmetic of Z[ξ] for a prime root of unity ξ and do not extend to other roots of unity. Recently, for any integral homology 3–sphere, Habiro [6] constructed a unified invariant whose evaluation at any root of unity coincides with the value of the Witten–Reshetikhin–Turaev invariant at that root. Habiro’s unified invariant is an element of the following ring Ẑ[q] := lim ←−− n Z[q] ((1− q)(1 − q2)...(1 − qn)) . Every element f(q) ∈ Ẑ[q] can be written as an infinite sum f(q) = ∑ k≥0 fk(q) (1 − q)(1 − q)...(1 − q), with fk(q) ∈ Z[q]. In particular, for a root of unity ξ, the evaluation evξ(f(q)) ∈ Z[ξ]. Thus, for any integral homology sphere M , τM (ξ) is an algebraic integer at any root of unity ξ. The fact that the unified invariant belongs to Ẑ[q] is stronger, than just integrality of τM (ξ). We will refer to it as “strong” integrality. In [2] Laplace transform method for constructing unified invariants of rational homology 3– spheres was developed. An application of this method is the following result in [3]. Theorem (Beliakova–Le). For every closed 3–manifold M and any root of unity ξ of odd order, τM (ξ) ∈ Z[ξ]. Strong integrality of quantum invariants for rational homology 3–spheres was studied in [10] and [3]. In [10], for a rational homology 3–sphere M with |H1(M,Z)| = b, Le constructed an invariant IM which dominates SO(3) quantum invariants of M at roots of unity of order coprime to b. Habiro’s universal ring was modified by inverting b and cyclotomic polynomials of order not coprime to b. In [3], it was proved that IM has even stronger integrality, i.e. it belongs to a smaller ring. Moreover, a rational surgery formula for this invariant was given. In this paper, we extend the theory to the case, where the orders of the root of unity and of the torsion are not coprime. Our method uses Andrews’s identities generalizing those of Rogers– Ramanujan. 1 2 ANNA BELIAKOVA, IRMGARD BÜHLER, AND THANG LE Results. Let Mb be the set of rational homology 3–spheres with |H1(M,Z)| = b. Let us fix an odd divisor c of b and let b = ∏n i=1 p ki i be the prime decomposition of b, i.e. the pi’s are distinct. We have H1(M ;Z) = n ⊕ i=1 mi ⊕ j=1 Z p kij i with ∑mi j=1 kij = ki. We renormalize τ(M) as τ ′ M (ξ) = τM (ξ) n ∏ i=1 mi ∏ j=1 τ L(p kij i ,1) (ξ) . We say that a closed 3–manifold is of diagonal type if it can be obtained by an integral surgery along an algebraically split link. Lemma 1. There are connected sums of lens spaces Modd and Mev with links inside, such that Modd is uniquely determined by M and M ′ := M#Modd#Mev is of diagonal type. Moreover, τ ′ Modd(ξ) is invertible in Z[ξ]. Suppose c = ∏ i p ki,c i is the prime decomposition of c. Define c ′ := ∏ i p ki,c i and b ′′ := ∏ i p ki−ki,c i , where the products are taken over i with 2ki,c < ki only. Put t := q c′ b′′ and b = b/c. We denote by Φi(x) the i–the cyclotomic polynomial in x and define a subring of Q(q ′′ ) by Rb,c := Z[q, t][Φ n (t) if (n, b) 6= 1, Φ j (q) if c ∤ j ] and let R̂b,c := lim ←−− k Rb,c ((q; q)k) be its cyclotomic completion, where (a; b)k = ∏k−1 i=0 (1 − ab). Let S = {r ∈ N | (r, b) = c}. For f ∈ R̂b,c and a root of unity ξ with ord(ξ) ∈ S, we define evξ(f) by sending q to ξ and t = q c/b to (ξ ′ ), where bd = 1 modulo ord(ξ)/c. Note that evξ(f) ∈ Z[1/b][ξ] in general. We single out a subring Γ̂b,c of R̂b,c, such that evξ(Γb,c) = Z[c][ξ], if b is odd, and evξ(Γb,c) = Z[(2c)][ξ] if b is even. We put bij = p kij i , cij = p kij,c i = (c, bij) and tij = q p 2kij,c−kij i . Notice, that tij are powers of t for all i and j. Any f ∈ Γ̂b,c admits the following presentation