BOOK REVIEW: AN EXTENSION OF CASSON’S INVARIANT BY KEVIN WALKER

REVIEWED BY: WILLIAM M. GOLDMAN

May 16, 1993

An integral (respectively rational) homology 3-sphere is a closed 3-manifold whose first integral (respectively rational) homology group vanishes. (We henceforth abbreviate these terms as ℤHS and ℚHS respectively.) These represent an important and slippery class of 3-manifolds — including, for example, those 3-manifolds for which Thurston’s tantalizing Geometrization Conjecture has yet defied proof. Furthermore such 3-manifolds are as plentiful as knots: Given any knot K ⊂ S^{3}, the various Dehn fillings obtained by attaching to the complement S^{3} − K solid tori are all homology spheres. Before 1985 the principal invariant of a ℤHS Σ^{3} was the ℤ/2-valued Rohlin invariant μ(Σ^{3}), derived from the signature of a bounding 4-manifold). At that time it seemed plausible that the Rohlin invariant could detect a possible counterexample to the Poincaré conjecture.

In 1985 Andrew Casson found a powerful ℤ-valued invariant λ(Σ^{3}) of a ℤHS Σ^{3}, which laid these hopes to rest. From its existence and basic properties follow some remarkable results in geometric topology, such as: there exist topological 4-manifolds which are not homeomorphic to simplicial complexes. A closely related result concerns the Poincaré conjecture: Casson’s invariant is defined in terms of the fundamental group and reduces modulo two to Rohlin’s invariant. In particular Rohlin’s invariant cannot detect a counterexample to the Poincaré conjecture.

(For an exposition of Casson’s original results, presented in three lectures at MSRI in Spring 1985, we recommend the notes [[1]] of Akbulut and McCarthy. For other generalizations similar to Walker’s invariant, see [4], [3], [2], [5], [7])

Shortly thereafter, Cliff Taubes found a highly suggestive analytic interpretation ([8]) of Casson’s λ(Σ^{3}). Let Σ^{3} be an oriented ℤHS and consider the space A of all connections on a fixed (trivial) SU(2)-bundle P over Σ^{3}. The group G of gauge transformations of P is isomorphic to the group of all maps from Σ^{3} into SO(3). Then the tangent space to A at A is the space Ω^{1}(Σ^{3}, 𝔤) of 1-forms on Σ^{3} taking values in the bundle of Lie algebras associated to P. The curvature F (A) of A is an exterior 2-form with values in 𝔤, an element of Ω^{2}(Σ^{3}, 𝔤). Using the invariant inner product on the Lie algebra of SU(2) and a volume form on Σ^{3}, the spaces T_{A}A = Ω^{1}(Σ^{3}, 𝔤) and Ω^{2}(Σ^{3}, 𝔤) are canonically dual. Thus the operation assigning to a connection A ∈ A its curvature F (A) ∈ Ω^{2}(Σ^{3}, 𝔤) is a 1-form Φ on A. Furthermore G acts on A preserving Φ. The zeroes of Φ evidently correspond to flat connections. Gorbits of flat connections correspond to equivalence classes of representations of the fundamental group into SU(2). Taubes defines a Poincaré-Hopf index of the 1-form Φ on A and shows that the resulting Euler characteristic equals Casson’s invariant.

1980 Mathematics Subject Classification (1985 Revision). Primary 57M05 (Low-dimensional topology).

Key words and phrases. homology sphere, representation space, Alexander polynomial, symplectic manifold .

This invariant depends on the Fredholm structure of the moduli space of flat SU(2)- connections on Σ^{3}. The “instanton homology” (see [6]) invented by A.Floer is a homology theory associated to Σ^{3} whose Euler characteristic equals λ(Σ^{3}). However, the Floer homology has been rather difficult to compute except in the simplest examples, while λ(Σ^{3}) can be computed simply in terms of a surgery formula discovered by Casson.

Given the power of this invariant, it was natural to try to extend it in various directions. The book under review (essentially the author’s Berkeley dissertation) describes an extension of λ(Σ^{3}) to the case when Σ^{3} is a ℚHS. If K ⊂ S^{3} is a knot, then its “complement” is a compact 3-manifold N^{3} with boundary a 2-torus (whose interior is homeomorphic to S^{3} − K). One obtains a rational homology sphere N_{p/q} by attaching to M^{3} a solid torus T ≈ S^{1} × D^{2} by glueing the meridian ({s} × ∂D^{2} of T to a (p, q)-curve on ∂N^{3}. (A (p, q)-curve on ∂N^{3} is defined as any curve in the homology class corresponding to (p, q) ∈ ℤ ⊕ ℤ ≅ H_{1}(∂N) which is the linear combination: p times a generator (1, 0) of the image of

ℤ ≅ H_{1}(N) ↪ H_{1}(∂N)

plus q times an element (0, 1) projecting to a generator under

H_{1}(∂N) ⟶ H_{1}(∂N)/H_{1}(N) ≅ ℤ

In particular N_{1/q} is a ℤHS for each q ∈ ℤ while every N_{p/q} is a ℚHS.

Casson’s original definition and Walker’s generalization involve both the moduli space Hom(π_{1}(M), SU(2))/SU(2) of equivalence classes of representations of the fundamental group π_{1}(M) into SU(2). The representations themselves form a real algebraic set upon which the inner automorphisms of SU(2) act algebraically. Accordingly the moduli space is a real analytic space, and stratifies into analytic submanifolds. Roughly, Casson’s invariant counts — algebraically — the points in Hom(π_{1}(M), SU(2))/SU(2) where M^{3} is a closed oriented ℤHS.

The actual counting is a bit more involved. Following Casson, one decomposes the 3-manifold Σ^{3} as a union of two solid handlebodies W_{1} ∪_{F} W_{2} of genus g ≥ 1 along a closed surface F and computes the space Hom(π_{1}(Σ^{3}), SU(2))/SU(2) in terms of this Heegaard decomposition. The space of interest Hom(π_{1}(Σ^{3}), SU(2))/SU(2) is embedded in the larger space Hom(π_{1}(F), SU(2))/SU(2) corresponding to the surface F ⊂ Σ^{3} under the natural restriction map, but there is considerably more space (and geometry) in Hom(π_{1}(F), SU(2))/SU(2) to study Hom(π_{1}(Σ^{3}), SU(2))/SU(2). Let X = Hom(π_{1}(F), SU(2))/SU(2). Then the space of equiavalence class of representations

Hom(π_{1}(Σ^{3}), SU(2))/SU(2) ⊂ X

(alternately, the space of gauge-equivalence classes of flat connections) is the intersection of the images Q_{i} of

Hom(π_{1}(W_{i}), SU(2))/SU(2) ↪ X

The orientation of Σ^{3} induces orientations of W_{1}, W_{2}, F ⊂ Σ^{3}, which in turn induce orientations on the various spaces Q_{1}, Q_{2} ⊂ X and Casson’s invariant λ(Σ^{3}) is a kind of oriented intersection number Q_{1} · Q_{2}.

To develop intuition for the nature of this invariant, consider the following simple analogue. Replace SU(2) by the circle group U(1). If Σ^{3} is a ℤHS, every representation π_{1}(Σ^{3}) ⟶ U(1) is trivial, but for a more general ℚHS, (say one with cyclic fundamental group ℤ_{m}), nontrivial representations π_{1}(Σ^{3}) ⟶ U(1) will exist. It suffices to consider the case that Σ^{3} is a lens space: glue two solid tori W_{1} and W_{2} along a common bounding 2-torus T. The inclusions of the boundary T ⟶ ∂W_{j} ⊂ W_{j} induce homomorphisms

ℤ ⊕ ℤ ≅ π_{1}(T) ⟶ π_{1}(W_{j}) ≅ ℤ
(x, y) ⟼ a_{j}x + b_{j}y

where a_{j} and b_{j} are relatively prime integers. Then the analogue X′ of X is the torus Hom(π_{1}(T), U(1)) dual to π_{1}(T). Explicitly X′ consists of representations of the form

ℤ ⊕ ℤ ⟶ U(1)
(x, y) ⟼ e^{i(αx+βy)}

where α, β ∈ ℝ. The analogues of the Q_{j} are the circles Q′_{j} (one-parameter subgroups with respect to the structure of X as a group) defined by

α = ta_{j}, β = tb_{j}

All of the intersections of Q′_{1} and Q′_{2} are transverse and have the same oriented intersection number. The cardinality of Q′_{1} ∩ Q′_{2} is the determinant m = a_{1}b_{2} − a_{2}b_{1}, the order of π_{1}(Σ^{3}). Thus #π_{1}(Σ^{3}) may be regarded as a “abelian” analogue of Casson’s invariant.

The book under review is a monograph describing Walker’s extension of Casson’s invariant to ℚHS. This was Walker’s 1989 Berkeley doctoral thesis and a summary of this work may be found in his research announcement [9]. The idea of Walker’s generalization is similar to Casson’s definition: sum local expressions over the intersections of Q_{1} and Q_{2} in X, after perhaps making a generic perturbation to make them transverse. However, the presence of singularities in these spaces produces significant technical problems. Using the symplectic geometry of X (in which the Q_{j} are Lagrangian), Walker defines local invariants which give a meaningful invariant. §1 is background on the moduli spaces of representations and its symplectic geometry. §2 contains the definition of λ(Σ^{3}) and §3 develops its elementary properties. Many of these were proved by Casson in ℤHS case. §4 proves the surgery formula: how λ(Σ^{3}) behaves when surgery is performed on a knot in a ℚHS. Conversely, uniqueness of Walker’s invariant is proved in §5: any function of rational homology spheres vanishing on S^{3} and satisfying the surgery formula must equal λ(Σ^{3}). §6 contains various consequences of this theory. For example Walker’s λ is additive under connected sum and relates to the Rohlin invariant of a spin ℚHS. Two appendices summarize results concerning Dedekind sums and Alexander polynomials which are needed in the main body of the monograph.

This is a fascinating subject and Walker’s book is informative and well written. Together with [1], it makes a rather pleasant introduction to a very active area in geometric topology.

References

1. Akbulut, S., and McCarthy, J., “Casson’s invariant for oriented homology 3-spheres: an exposition,” Mathematical Notes 36, Princeton University Press, Princeton, New Jersey (1990)

2. Boyer, S. and Lines,D., Surgery formulae for Casson’s invariant and extensions to homology lens spaces, UQAM Rapport de recherche 66 (1988)

3. Boyer, S. and Nicas, A., Varietes of group representations and Casson’s invariant for rational homology 3-spheres, preprint (1986)

4. Cappell, S., Lee,R., and Miller,E.Y., A symplectic geometry approach to generalized Casson’s invariants of 3-manifolds, Bull. A. M. S. 22 (2), (1990), 269–275

5. Curtis-Budka, C.L., Casson-type invariants counting representations in low-rank groups, Doctoral thesis, Yale University (1991)

6. Floer, A., An instanton-invariant for 3-manifolds, Comm. Math. Phys. 118 (21), (1988), 5–240 7. Lescop, C.,

8. Taubes,C.H., Casson’s invariant and gauge theory, J. Diff. Geo. 31 (1990), 547–599

9. Walker, K., An extension of Casson’s invariants to rational homology spheres, Bull. A. M. S. 22 (2), (1990), 261–267

Department of Mathematics and Institute of Advanced Computer Studies University of Maryland, College Park, MD 20742

E-mail: wmg@math.umd.edu