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COHOMOLOGY RINGS, LINKING FORMS AND INVARIANTS OF SPIN STRUCTURES OF THREE-DIMENSIONAL MANIFOLDS

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1984 Math. USSR Sb. 48 65 (http://iopscience.iop.org/0025-5734/48/1/A04) View the table of contents for this issue, or go to the journal homepage for more

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MaxeM. C5opHHK TOM 120(162) (1983), Bun. 1

Math. USSR Sbornik Vol. 48(1984), No. 1

COHOMOLOGY RINGS, LINKING FORMS AND INVARIANTS OF SPIN STRUCTURES OF THREE-DIMENSIONAL MANIFOLDS UDC 513.836 V. G. TURAEV ABSTRACT. Necessary and sufficient conditions on a triple consisting of a sequence of graded rings, a bilinear form, and a function with values in Z/16 are given which ensure that the triple consists of the cohomology rings, the linking form, and the Rokhlin function of some closed oriented three-dimensional manifold. Bibliography: 24 titles. Figures: 2.

§1. Introduction

1.1. Object of the paper. The cohomology ring and the linking form are fundamental (and classical) combinatorial invariants of a manifold. The Rokhlin function also plays an important role in the topology of three-dimensional manifolds—it is the natural generalization of the Rokhlin invariant of a homology sphere to an arbitrary closed orientable three-dimensional manifold Μ and is a map from the set of spin structures on the tangent bundle of Μ to Z/16 (see [6], [9] or §1.3). In what follows, we let Jt denote the set of closed orientable three-dimensional manifolds. In this paper, we investigate the algebraic relationships which exist among the Rokhlin functions, the linking forms, and the cohomology rings with coefficients in Ζ/η, η = 0,1,2,..., of manifolds in Jt'. In particular, we establish a number of new relations among these invariants. The most interesting of these is a formula showing that the cohomology ring with coefficients in Z/2 of a manifold in J( is determined by its Rokhlin function (see §1.3). The premise that such an explicit formula existed served as the starting point of this paper. This premise was, to a considerable extent, predicated upon a desire to carry over to manifolds the Kaplan formula [9] expressing the triple Milnor number of links in terms of the Arf invariant of knots. We also establish a formula which shows that the squaring operation Hl(M;Z/n) -> H2(M\Z/n), where Μ &.Jt and η > 0, is determined by the linking form of Μ (see §1.2). Despite the simplicity of this formula and the fact that it relates the classical invariants of a manifold, the author has not succeeded in finding it in the literature. Yet another relation among the invariants we consider is that the values of the Rokhlin function modulo 8 can be computed using the linking form (see §1.4). In a number of 1980 Mathematics Subject Classification. Primary 57M1O, 57R15; Secondary 57M25, 57R15, 57N65. '"1984 American Mathematical Society 0025-5734/84 $1.00 4- $.25 per pa»o 65

66

V. G TURAEV

cases (for example, for Z/2-homology spheres and in a number of related situations), this fact was known (see [3], [6] and [13]). However, our definitive formulation of the appropriate result is, apparently, new. Our main result is that the relations enumerated above exhaust all algebraic relations among the cohomology rings, the linking forms and the Rokhlin functions of a manifold inJf. We use this to formulate algebraic conditions on a triple (sequence of graded rings, bilinear form, function with values in Z/16) which are necessary and sufficient that its members be realized as the cohomology rings, the linking form, and the Rokhlin function of a manifold in Jt'. It should be noted that the question of which algebraic properties characterize the cohomology rings and linking forms of manifolds in Ji was studied earler. By the universal coefficient formula and Poincare duality, the cohomology rings of Μ e Ji are determined by the group HX{M) ( = HX{M\Z)) and the sequence of skew-symmetric trilinear forms

{(*, y, z) -» (x Uy U z)([M]): {Hl{M;Z/n)f

-

Z/n}n>0,

henceforth denoted by un{M), where η > 0. (The form ux(M) is clearly trivial, but it will be convenient not to exclude it from consideration.) The problem of characterizing the forms un algebraically was first considered by Postnikov [19] in 1942 for the case η = 2. He proved that if Η = (Z/2) r for some 0 < r < oo, then a (skew) symmetric trilinear form u: H3 -» Z/2 is isomorphic to u2(M) for some Μ e ,#if and only if u(x, x, y) = u(y, y, x) for all x, y e H. (Postnikov obtained a similar result for nonorientable manifolds; the nonorientable case will not be considered here.) The case of integral cohomology was studied by Sullivan [21], who proved that an arbitrary skew-symmetric trilinear form (Z r ) 3 -» Ζ for 0 < r < oo is isomorphic to uo(M) for some Μ eJ/. Concerning linking forms, Kawauchi and Kojima recently proved [10] that if Γ is a finite abelian group, then an arbitrary nondegenerate symmetric bilinear form Τ Χ Τ -» Q/Z can be realized as the linking form (Tors HX{M))2 -> Q/Z of a manifold Μ e Jf. The question of characterizing Rokhlin functions algebraically has not, as far as the author knows, previously been considered. The main result of this paper is formulated in §1.5. In §§1.2-1.4, we introduce the notation we need and formulate some intermediate results concerning the algebraic characterization of the invariants under consideration. Our terminology is that of differential topology. 1.2. The cohomology rings and linking forms of manifolds inJf. If Η is an abelian group and η ^ 0, then H* will denote the group Hom(//, Z/n). A sequence of forms {un: (Η*γ -> Z/n}n>0 is called compatible if for any m, n, n 1 ; . . . ,n4 > 0, and for any ?j, e Hom(H, Z/n,) with / = 1,2,3, and for any commutative diagram Ζ/ηλχΖ/η2 a, X « 2 X a 3 /

\

Χ Ζ/η3 β, Χ β2 Χ β3

{Z/mf i a4

I ft,

Z/m -»Z/n4 «- Z/n

SPIN STRUCTURES ON THREE-DIMENSIONAL MANIFOLDS

67

in which the vertical arrows correspond to the maps induced by multiplication on the quotient rings and ax,...,a4 and β1,...,β4 are linear (not necessarily ring) homomorphisms, the following identity holds: «4( a ™( ( J ril> a 2 < ' T l2' a 3° I i3)) = A ( ! ' » ( A ' ' 1 l . f t ° i 2 ^ 3 o i 3 ) ) · The compatibility condition appears quite cumbersome. However, it reflects the fact that homomorphisms of the coefficients serve to relate the forms {un(M)}n>0, Μ e Μ, not only to one another, but also to the maps Hl{M\Z/nx)

X Hl{M\Z/n2)

X Hl{M; Z/n 3 ) -> Z/n 4

corresponding to the various trilinear forms Z/nx X Z/n 2 X Z/n 3 -* Z/n 4 . (In fact, these maps can be recovered from {un(M)}n>0.) It is easy to verify that {u n (M)} n > 0 is compatible in the above sense. The linking form of a manifold Μ e. Ji will be denoted by L(M). We identify the groups Hl(M; Z/n) and Hom(//1(M), Z/n) in the usual manner. I. Let Η be a finitely generated abelian group, let L: (Tors H)2 -» Q/Z be a nondegenerate symmetric bilinear form, and let {«„: (Η*Ϋ -» Z/n}n>0 be a compatible sequence of skew-symmetric trilinear forms. In order that there exist a manifold Μ e Ji and an isomorphism Η —> Ηλ{Μ) sending L to L(M) and un to un(M) for all η it is necessary and sufficient that the following condition (*)„ be satisfied for all even η > 2. (*)„. //ψ w the inclusion k (mod n)1-* k/n; Z/n -» Q/Z, then THEOREM

φ(ι*η(χ,χ,γ))

= ~L(x, y)

for all x,y e //*; here χ andy are elements of the group Tors Η for which L(x, α) = ψ(χ(α)) and L(y, a) = 4>{y(a))for all a e Tors H. The necessity of condition (*)„ follows from the fact that for even η > 2 the squaring operation H1(M; Z/n) -* H2(M; Z/n) is a multiple of the Bockstein homomorphism β: Hl(M; Z/n) -» H2(M; Z/n). More precisely, if χ e H\M; Z/n), then χ2 = (η/2)β(χ). (This equality holds on the cohomology of any cell-like space; it is easily verified in H*(K(Z/n,l); Z/n).) The duality homomorphism H2(M;Z/n) -> H\M;Z/n) takes to x. Thus, for any x, y e Hl(M; Z/n), we have

Although it will not be necessary in what follows, we remark that one can limit oneself to a finite number of conditions (*)„ in the formulation of Theorem I. If 2N is the order of a maximal cyclic 2-subgroup of Tors H, then it is easy to verify by purely algebraic means that the conditions (*)„, for any even n, follow from the conditions (*)2< with / = 1,... ,N. THEOREM II. Let Η be a finitely generated abelian group and let u: (Η*Ϋ —> Z/« be a skew-symmetric trilinear form where η > 0. // η = 0 or if η is odd, then there is an Μ 2, then there exists an Μ e.Ji such that the pair (H,u) is isomorphic to {Hl(M),un(M)) if and only if there exists a nondegenerate symmetric bilinear form L: (Tors H)2 -* Q/Z such that condition (*)„ is satisfied with un = u.

68

V.G.TURAEV

When η > 2 is even, the condition formulated in Theorem II implies that un(x, x, y) = un{y, y, x) for all x j e H*, but does not reduce to this. For example, if Η = Z r , then the existence of the required form L is equivalent to u(x, x, y) = 0 for all i j e H*. If Η = (Z/2) r and « = 2, then the existence of L is equivalent to the conditions that u(x, x, y) = u(y, y, x) for all x, y e H* and that for each χ e tf* there exists y e //* with w(x, x, y) Φ 0. The following sharper form of Postnikov's mentioned-above result follows from Theorem II: if Η = (Z/2) r , where 0 «S r < oo and «: // 3 -> Z/2 is a (skew) symmetric trilinear form with u(x, x, y) = u(y, y, x) for all x, y e //, then there exists Μ ^Jt such that u is isomorphic to « 2 (M) and Ηλ(Μ) = (Ζ/4)5 Θ (Z/2)r~·5, where s is the dimension of the space {x e H: u(x Χ Η Χ //) = 0}. 1.3. 77ze Rokhlin function and the mod 2 cohomology ring. Let spin(M) denote the set of spin structures on the manifold Μ e J( and let R(M) denote the Rokhlin function spin(M) -> Z/16. Recall that spin(M) is canonically equipped with the structure of an affine Hl(M; Z/2)-space (see [16]). To a spin structure α on Μ the Rokhlin function associates the residue mod 16 of the signature of V, where V is a compact four-dimensional manifold with boundary Μ which admits a spin structure extending a (see [6] and [9])· THEOREM III. // Μ e J(, R = R(M) and u-, = M 2 ( M ) , then for any a e spin(M) and any χγ, x2, x 3 e //^M; Z/2)

= «(a) -

Su2{xl,x2,x-i)

Σ

Λ(α + χ,.)

1 Z/16. If the manifold W, introduced in the proof of Lemma 2.2, is endowed with the orientation extending the orientation on V, then the additivity of the signature implies that the left-hand side of (6) is equal to the signature of W (mod 16). By the generalized Rokhlin theorem, the latter residue is equal to PROOF.

[ F ] [ F ] + 2B(a F ) = 2B(otF). 2.4. LEMMA. // q: Z/2 -» Q/Z is the quadratic form associated with a nontrivial bilinear pairing (Z/2) 2 -» Q/Z, then either q{\) = \ and B(^f) = Imod8 or q{\) = - \ and B(?) = -Imod8. If q: (Zl/2)el © (Z/2)e 2 ~* Q/Z is the quadratic form associated with the bilinear pairing exex = e2e2 = 0, exe2 = e2el = \, then