Rohlin?s invariant and gauge theory, I. Homology 3-tori
Commentarii Mathematici Helvetici
This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U (2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y . Our counting argument makes use of a natural action of H 1 (Y ; Z 2 ) on the moduli
... Z 2 ) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Casson's homology sphere invariant reduces mod 2 to the Rohlin invariant. Rohlin's invariant and gauge theory, I 619 manifolds obtained by surgery on a knot. Our main contribution is to identify a Casson-type invariant of a homology 3-torus, Y , with a Rohlin-type invariant. This is accomplished by relating the action of the group H 1 (Y ; Z 2 ) on the moduli space of (projectively) flat connections to the cup product in cohomology. The techniques we develop to deal with equivariant aspects of non-smooth moduli spaces should be of independent interest in the study of instanton Floer homology (compare [7, §5.6]). We have made use of the perturbation theory and other techniques from this paper in subsequent work [17, 18] . Let us briefly describe the invariants in question; more details will be given in the next section. By homology 3-torus we mean a closed oriented 3-manifold Y having the integral homology of the 3-torus T 3 = S 1 × S 1 × S 1 . For any nontrivial w ∈ H 2 (Y ; Z 2 ), we consider projectively flat connections on a principal U (2)-bundle P → Y whose associated SO(3) = P U(2)-bundle has second Stiefel-Whitney class equal to w. We define a Casson-type invariant λ (Y, w) to be one-half of the signed count of such connections, modulo an appropriate gauge group. This invariant is one-half of the Euler characteristic of the Floer homology studied in  and  and is not, a priori, an integer. A pair consisting of a closed oriented 3-manifold X and a spin structure σ has a Rohlin invariant ρ(X, σ) ∈ Q/2Z. By definition, for any spin 4-manifold V with (spin) boundary (X, σ). By the Rohlin invariant ρ (Y ) smooth compact of a homology 3-torus Y we mean the sum, over the eight spin structures on Y , of their Rohlin invariants. It is easy to see that, as for a homology sphere, this invariant actually takes values in Z/2Z. Theorem 1.1. For any choice of non-trivial Note that this implies that λ (Y, w) (mod 2) is independent of w, so long as w is non-trivial. It is a theorem of V. Turaev  , based on S. Kaplan [12, Lemma 6.3] that the above triple cup product also evaluates the Rohlin invariant. Hence we obtain the following result. At the end of the paper, we will explain how this implies Casson's original result about the Rohlin invariant of homology spheres. We conjecture that the congruence in Theorem 1.1 lifts to the integers as λ (Y, w) = ±((a 1 ∪ a 2 ∪ a 3 ) [Y ]) 2 . This is suggested by Casson's original work [1, 19] , and a closely related formula is given by Lescop . To prove this conjecture, one would have to show that the count of flat connections on a homology 3-torus 620 D. Ruberman and N. Saveliev CMH equals either Lescop's invariant or Casson's invariant for 3-component links (of trivial linking numbers). This equality is closely related to Casson's formula for his knot invariant in terms of the Alexander polynomial; a purely gauge-theoretic proof of the latter has been given by Donaldson . The techniques in the present paper are rather different, and moreover have the advantage of extending to the 4-dimensional situation  where the integral version does not hold. The idea of the proof of Theorem 1.1 is to take advantage of a natural H 1 (Y ; Z 2 ) = (Z 2 ) 3 action on the moduli space of projectively flat connections. We identify this moduli space with the space of projective representations of π 1 Y in SU (2), and use this identification to show that the orbits with two elements are always nondegenerate and that the number of such orbits equals (a 1 ∪ a 2 ∪ a 3 ) [Y ] (mod 2). In the non-degenerate situation, this completes the proof because there are no orbits with just one element, and the orbits with four and eight elements do not contribute to λ (Y, w) (mod 2). The general case reduces to the non-degenerate one after one finds a generic perturbation which is H 1 (Y ; Z 2 )-equivariant. As mentioned above, this equivariance is rather delicate. The authors thank Christopher Herald for sharing his expertise. Vol. 79 (2004) Rohlin's invariant and gauge theory, I 623 Proposition 2.2. The action of H 1 (Y ; Z 2 ) preserves the mod 2 Floer index. Proof. This follows from [3, pages 239-240]. Remark 2.3. According to Proposition 2.2, the points in the H 1 (Y ; Z 2 )-orbit of a projectively flat connection A are counted in λ (Y, w) with the same sign. Hence we could as well define λ (Y, w) by counting points in M(P ), where w 2 (P ) = w, with weights given by the order of the orbits of their respective lifts to M(P ). Projective representations The holonomy map gives a homeomorphism between the moduli space M(P ) of flat connections onP and the SO(3)-character variety of π 1 Y . Similarly, there is an algebraic interpretation (again using holonomy) of projectively flat connections in terms of projective representations. This section describes this concept in some detail; good general references for these ideas are the classic paper of Atiyah-Bott  and the book of Brown . 3.