FLOER HOMOLOGY OF BRIESKORN HOMOLOGY SPHERES : SOLUTION TO ATIYAH’S PROBLEM

NIKOLAI SAVELIEV

Abstract. In this paper we answer the question posed by M. Atiyah, see [12], and give an explicit formula for Floer homology of Brieskorn homology spheres in terms of their branching sets over the 3–sphere. We further show how Floer homology is related to other invariants of knots and 3–manifolds, among which are the μ̄–invariant of W. Neumann and L. Siebenmann and the Jones polynomial. Essential progress is made in proving the homology cobordism invariance of our own ν–invariant, see [15].

Let p, q, and r be pairwise coprime positive integers. The Brieskorn homology 3–sphere Σ(p, q, r) is the link of the singularity of f^{−1}(0), where f : ℂ^{3} → ℂ is a map of the form f(x, y, z) = x^{p} + y^{q} + z^{r}. The complex conjugation in ℂ^{3} acts on Σ(p, q, r) turning it into a double branched cover of S^{3} branched over a Montesinos knot k(p, q, r). In this paper, we express the Floer instanton homology groups I_{n}(Σ(p, q, r)), see [5], in terms of the Casson’s λ–invariant of Σ(p, q, r) and the signature of the knot k(p, q, r).

Theorem 1. The Floer homology group I_{n}(Σ(p, q, r)) is trivial if n is odd, and is a free abelian group of the rank

rank I_{n}(Σ(p, q, r)) = \frac{1}{16} ( 8λ(Σ(p, q, r)) − (−1)^{n/2} sign k(p, q, r))

if n is even.

The groups I_{∗}(Σ(p, q, r)) were studied by R. Fintushel and R. Stern in [6] (we use the Floer index convention of this paper). They gave an algorithm which for any given set of positive coprime integers p, q, r computes the ranks of I_{∗}(Σ(p, q, r)). However, their approach does not give a closed form formula for I_{∗}(Σ(p, q, r)). It should be mentioned though that the vanishing of the groups I_{n}(Σ(p, q, r)) for odd n follows from [6]; we give an alternative proof in Section 3.

1991 Mathematics Subject Classification. 57M25, 57R80.

Key words and phrases. Floer homology, Seifert manifolds, Montesinos knots, Casson invariant, Jones polynomial, homology cobordism, knot signature.

Research at MSRI is supported in part by NSF grant DMS-9022140.

In [15] we introduced an invariant ν, which for Brieskorn homology spheres takes the form

ν(Σ(p, q, r)) = \frac{1}{2} ∑_{n=0}^{3}(−1)^{n+1} rank I_{2n}(Σ(p, q, r)).  (1)

Note that if the ranks in (1) were added without plus or minus signs one would get the Casson invariant λ(Σ(p, q, r)). We conjectured in [15] that the ν–invariant equals the μ̄–invariant of W. Neumann [13] and L. Siebenmann [17] for all Seifert fibered homology spheres. We prove that our conjecture is true. The first step is given by the following theorem.

Theorem 2. For any Brieskorn homology 3–sphere Σ(p, q, r), the ν–invariant defined by (1) and the μ̄–invariant coincide, ν(Σ(p, q, r)) = μ̄(Σ(p, q, r)).

It can be easily seen that Theorem 2 is in fact equivalent to Theorem 1 due to the following two observations: first, μ̄(Σ(p, q, r)) = 1/8 · sign k(p, q, r), see [17], and second, I_{n}(Σ(p, q, r)) = I_{n+4}(Σ(p, q, r)) for all n. Second step in proving our conjecture makes use of the so called splicing additivity proven for both the μ̄– and ν–invariants in [15]. Namely, μ̄(Σ(a_{1}, …, a_{n})) = μ̄(Σ(a_{1}, …, a_{j}, p))+μ̄(Σ(q, a_{j+1}, …, a_{n})), where j is any integer between 2 and n−2, and the integers q = a_{1} ⋯ a_{j} and p = a_{j+1} ⋯ a_{n} are the products of the first j and the last (n − j) Seifert invariants, respectively. The same additivity holds for the invariant ν.

Theorem 3. For any Seifert fibered homology 3–sphere Σ(a_{1}, …, a_{n}), the invariants ν and μ̄ coincide, ν(Σ(a_{1}, …, a_{n})) = μ̄(Σ(a_{1}, …, a_{n})).

Another reformulation of Theorem 1 establishes links between Floer homology and Jones polynomial. The result below follows from Theorem 1 and D. Mullins formula for the Casson invariant of two-fold branched covers [11].

Theorem 4. Let Σ(p, q, r) be a Brieskorn homology sphere then

rank I_{0}(Σ(p, q, r)) = −\frac{1}{12} · \frac{d}{dt}|_{t=−1} ln V_{k} (t)

where V_{k} is the Jones polynomial of the Montesinos knot k = k(p, q, r).

Theorem 3 has the following application to the homology 4–cobordism. In [15] and [16] we proved that μ̄ is a homology cobordism invariant for some classes of Seifert fibered homology 3–spheres. The corresponding results for the ν–invariant hold due to the identification μ̄ = ν in Theorem 3.

Theorem 5. (1) Let Σ(a_{1}, …, a_{n}) be a Seifert fibered homology sphere homology cobordant to zero. Then ν(Σ(a_{1}, …, a_{n})) ≥ 0.

(2) Let Σ = Σ(p, q, pqm ± 1) be a surgery on a (p, q)–torus knot, and suppose that Σ is homology cobordant to zero. Then ν(Σ) = 0.

(3) For all known Seifert fibered homology 3–spheres Σ which are known to be homology cobordant to zero including the lists of Casson-Harer [3] and Stern [18], ν(Σ) = 0.

We will prove that ν(Σ(p, q, r)) = 1/8·sign k(p, q, r), which is equivalent to proving Theorem 1. The idea is shortly as follows. In [19], C. Taubes proved that, for any homology 3–sphere Σ,

λ(Σ) = 1/2 · χ(I_{∗}(Σ)).  (2)

The Casson’s λ–invariant on the left is defined topologically using a Heegaard splitting of Σ and SU(2)–representation spaces. The number on the right is the Euler characteristic of Floer homology I_{∗}(Σ); it can be defined using SU(2) gauge theory as an infinite dimensional generalization of the classical Euler characteristic.

Let now Σ = Σ(p, q, r) be endowed with the involution σ induced on Σ ⊂ ℂ^{3} by the complex conjugation. We work out σ–invariant versions of both invariants in (2) in this particular situation. We use a σ–invariant Heegaard splitting of Σ and the corresponding σ–invariant SU(2)–representation spaces to define an invariant λ^{ρ}(Σ) in a manner similar to that of A. Casson. On the other hand, a σ–invariant gauge theory produces a σ–invariant Euler characteristic which we denote χ^{ρ}(Σ). Note that the latter can be defined without actually working out any Floer homology. Then, a σ–invariant version of the Taubes result (2) is that

λ^{ρ}(Σ(p, q, r)) = 1/2 · χ^{ρ}(Σ(p, q, r)).  (3)

Our next step is to show that λ^{ρ}(Σ(p, q, r)) = 1/8 · sign k(p, q, r). We achieve this by pushing σ–invariant representations down to the quotients. We do this for all the manifolds in a σ–invariant Heegaard splitting and note that the push-down representations are only defined on the complements of the fixed point sets, and they map all the meridians to trace-free matrices in SU(2). After that we are in position to identify our λ^{ρ}–invariant with the Casson–Lin invariant 1/8 · sign k(p, q, r) of [8]. The crucial observation is that all the representations of π_{1}(Σ(p, q, r)) are in fact σ–invariant, and the λ^{ρ}–invariant just counts them with signs different from those defined by Casson for his λ–invariant.

Finally, we identify 1/2·χ^{ρ}(Σ) with ν(Σ) as follows. We choose a σ–invariant metric on Σ and an almost complex structure on Σ×ℝ so that the involution σ ×1 on Σ×ℝ is anti-holomorphic. At the level of tangent spaces, the ±1–eigenspaces of σ × 1 are then isomorphic to each other and hence have the same dimension. Therefore, the spectral flow used in [19] to define χ(Σ) splits in halves, and the ν–invariant becomes an “honest” σ–invariant Euler characteristic equal to 1/2 · χ^{ρ}(Σ).

The paper begins with an introduction to the relevant topology of Brieskorn homology spheres and Montesinos knots. In Section 2 the invariant λ^{ρ}(Σ(p, q, r)) is introduced and the equality λ^{ρ}(Σ(p, q, r)) = 1/8·sign k(p, q, r) is proven. The invariant χ^{ρ}(Σ(p, q, r)) is defined in Section 3; the equality 1/2 · χ^{ρ}(Σ(p, q, r)) = ν(Σ(p, q, r)) is proven in the same section. Section 4 is devoted to the proof of the fact that λ^{ρ}(Σ(p, q, r)) = 1/2 · χ^{ρ}(Σ(p, q, r)). Each section has its own introduction and is further subdivided.

It should be pointed out that most of the results in the paper hold in a more general situation. A description of the relevant results will appear elsewhere, as will a more detailed version of this preprint.

1. Topology of Brieskorn homology spheres

Let p, q, r be relatively prime positive integers greater than or equal to 2. The Brieskorn homology sphere Σ(p, q, r) is defined as the algebraic link

Σ(p, q, r) = { (x, y, z) ∈ ℂ^{3} | x^{p} + y^{q} + z^{r} = 0 } ∩ S_{1}^{5},

where S_{1}^{5} is the unit sphere in ℂ^{3}. This is a smooth naturally oriented 3–manifold with H_{∗}(Σ(p, q, r)) = H_{∗}(S^{3}). Moreover, Σ(p, q, r) is Seifert fibered, see [14], with the Seifert invariants { b, (p, b_{1}), (q, b_{2}), (r, b_{3}) } such that

b_{1}qr + b_{2}pr + b_{3}pq + bpqr = 1.  (4)

The complex conjugation on ℂ^{3} obviously acts on Σ(p, q, r) as

σ : Σ(p, q, r) → Σ(p, q, r), (x, y, z) ↦ (x̄, ȳ, z̄).  (5)

The fixed point set of this action is never empty. The quotient of Σ(p, q, r) by the involution σ is S^{3}, with the branching set the so called Montesinos knot k(p, q, r), see [10], [2], or [17]. The knot k(p, q, r) can be described by the following diagram

Figure 1 where a box with α, β in it stands for a rational (α, β)–tangle, see [2], Fig. 12.9. The parameters b, (p, b_{1}), (q, b_{2}), (r, b_{3}) are the Seifert invariants of the corresponding Σ(p, q, r). According to [2], Theorem 12.28, these parameters together with (4) determine the knot k(p, q, r) uniquely up to isotopy.

2. The invariant λ^{ρ}

In this section we first shortly recall the definition of Casson’s λ–invariant. Our λ^{ρ}–invariant for Σ(p, q, r) will be a σ–invariant version of λ. To define it, we introduce a σ–invariant Heegaard splitting of Σ(p, q, r). Then we define the corresponding σ–invariant representation spaces and investigate closely the representation space of π_{1}(Σ(p, q, r)). It turns out that all the representations in the latter space are σ–invariant. After computing the necessary dimensions and checking the transversality condition, we define the λ^{ρ}–invariant as an intersection number of σ–invariant representation spaces. Finally, we check that λ^{ρ}(Σ(p, q, r)) = 1/8 · sign k(p, q, r).

1. Casson invariant. Let M be an oriented homology 3–sphere with a Heegard splitting M = M_{1} ∪ M_{2} where M_{1} and M_{2} are handlebodies of genus g ≥ 2 glued along their common boundary, a Riemann surface M_{0}. Let

R(M_{i}) = Hom^{∗}(π_{1}(M_{i}), SU(2))/ ad SU(2), i = 0, 1, 2, ∅,

be the set of conjugacy classes of irreducible representations of π_{1}(M_{i}) is SU(2). Each R(M_{i}), i = 0, 1, 2, is naturally an oriented manifold. The dimension of R(M_{1}) and R(M_{2}) is 3g−3, and that of R(M_{0}) is 6g−6. The inclusions M_{0} ⊂ M_{i}, i = 1, 2, induce embeddings R(M_{i}) ⊂ R(M_{0}). The points of intersection of R(M_{1}) with R(M_{2}) are in one-to-one correspondence with R(M). If the intersection is transversal we define the Casson’s λ–invariant as

λ(M) = \frac{1}{2} ∑_{α∈R(M)} ε_{α}

where ε_{α} = ±1 is a sign obtained by comparing the orientations on T_{α}R(M_{1}) ⊕ T_{α}R(M_{2}) and T_{α}R(M_{0}). Note that the intersection is transversal if Σ = Σ(p, q, r).

2. The invariant Heegaard splitting. We are going to construct an invariant Heegaard splitting of Σ(p, q, r). Let us first fix notations. By Σ we denote a Brieskorn homology sphere Σ(p, q, r) and by σ : Σ → Σ the involution constructed in (5). The projection on the quotient space will be denoted by π, so π : Σ → Σ/σ = S^{3}. The projection π maps the fixed point set Fix(σ) ⊂ Σ onto the Montesinos knot k = k(p, q, r) ⊂ S^{3}.

The knot k ⊂ S^{3} can be represented as the closure of a braid β on n strings. We think about k as consisting of the braid β and n untangled arcs in S^{3} forming its closure, see Figure 2. Let S ⊂ S^{3} be an embedded 2–sphere in S^{3} splitting S^{3} in two 3–balls, B_{1} and B_{2}, with common boundary S, and such that the entire braid β belongs to int B_{1}. The intersection k ∩ S consists of 2n points P_{1}, …, P_{2n}.

Figure 2

Now, we define a Heegaard splitting Σ = M_{1} ∪_{M_{0}} M_{2} as follows:

M_{1} = π^{−1}(B_{1}), M_{2} = π^{−1}(B_{2}), M_{0} = π^{−1}(S).  (6)

Obviously, M_{1} and M_{2} are handlebodies branched over the braid β and the n untangled arcs, respectively. Their common boundary is M_{0}, which is a closed surface branched over the points P_{1}, …, P_{2n} ∈ S. The genus g of M_{0} can be figured out by comparing Euler characteristics,

2 · χ(S) = χ(M_{0}) + χ( Fix(σ) ∩ S),

where χ(S) = 2, χ( Fix(σ) ∩ S) = 2n, and χ(M_{0}) = 2 − 2g. Therefore, g = n − 1. The Heegaard splitting we just defined is σ–invariant in the sense tnat σ(M_{i}) = M_{i} for i = 0, 1, 2.

3. Representations of Brieskorn homology spheres. Let Σ(p, q, r) be a Brieskorn homology 3–sphere, and

R(Σ(p, q, r)) = Hom^{∗}(π_{1}(Σ(p, q, r)), SU(2))/ ad SU(2)  (7)

the space of the conjugacy classes of irreducible representations of its fundamental group in SU(2). In this subsection we are concerned with describing the space R(Σ(p, q, r)) and the involution σ^{∗} induced on the representation space by σ.

We will follow [6] and first identify the group SU(2) with the group S^{3} of unit quaternions in the usual way, so that

i = (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) , j = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , k = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) .

Under this identification, the trace tr(A) of an element A ∈ SU(2) coincides with the number 2 Re A, A ∈ S^{3}, and the mapping r : S^{3} → [0, π], r(A) = arccos(Re A), is a complete invariant of the conjugacy class of the element A. This conjugacy class is a copy of S^{2} in S^{3} unless A = ±1, in which case the conjugacy class consists of just one point.

The fundamental group π_{1}(Σ(p, q, r)) has the following presentation, see [2],

π_{1}(Σ(p, q, r)) = ⟨x, y, z, h | h central , x^{p} = h^{−b_{1}}, y^{q} = h^{−b_{2}}, z^{r} = h^{−b_{3}}, xyz = h^{−b}⟩.

Specifying an irreducible representation α : π_{1}(Σ(p, q, r)) → SU(2) amounts to specifying a set {α(h), α(x), α(y), α(z)} of unit quaternions. In fact, we only need to specify the first three quaternions in this set because α(z) will then be expressed in their terms as α(z) = (α(x)α(y))^{−1}α(h)^{−b}. Since h is central and the representation α is irreducible, α(h) = ±1. Let us denote ε_{i} = α(h)^{b_{i}} = ±1, i = 1, 2, and ε_{3} = α(h)^{b_{3}−rb} = ±1. Then the relations x^{p} = h^{−b_{1}}, y^{q} = h^{−b_{2}} and (xy)^{r} = h^{b_{3}−rb} imply the following restrictions on α(x) and α(y):

r(α(x)) = πℓ_{1}/p, r(α(y)) = πℓ_{2}/q, r(α(x)α(y)) = πℓ_{3}/r,

where ℓ_{i} is even if ε_{i} = 1, ℓ_{i} is odd if ε_{i} = −1, and 0 < ℓ_{1} < p, 0 < ℓ_{2} < q, 0 < ℓ_{3} < r.

After conjugation, we may assume that α(x) = e^{πiℓ_{1}/p}. The quaternions α(y) and α(x)α(y) should lie in their respective conjugacy classes, S_{2} = r^{−1}(πℓ_{2}/q) and S^{3} = r^{−1}(πℓ_{3}/r). On the other hand, α(x)α(y) lies in α(x) · S_{2}, therefore, in order for α(x) and α(y) to define a representation, the intersection α(x)·S_{2} ∩S_{3} must be non-empty. Since α(x) · S_{2} is a 2-sphere centered at α(x), the intersection α(x) · S_{2} ∩ S_{3} in S^{3} (if non-empty) is a circle. This circle parametrizes a whole collection of representations α′ coming together with α such that r(α′(x)) = r(α(x)), r(α′(y)) = r(α(y)), and r(α′(x)α′(y)) = r(α(x)α(y)). In fact, all these representations are conjugate to each other by simultaneous conjugation of α(x) and α(y) by the complex circle S^{1} ⊂ S^{3}. This can be seen from the following technical lemma.

Lemma 6. Let α and β be irreducible representations of the group π_{1}(Σ(p, q, r)) in SU(2) such that

(1) α(h) = β(h) and α(x) = β(x) ∈ ℂ,

(2) r(α(y)) = r(β(y)), and

(3) r(α(x)α(y)) = r(β(x)β(y)). Then the representations α and β are conjugate to each other, that is there exists a unit quaternion c such that

β(t) = c · α(t) · c^{−1} for all t ∈ π_{1}(Σ(p, q, r)).

Moreover, the quaternion c may be chosen to be a complex number, c ∈ ℂ.

Corollary 7. ( Compare [6] ). The representation space R(Σ(p, q, r)) is finite.

The involution σ : Σ(p, q, r) → Σ(p, q, r) induces the involution on the fundamental group,

σ_{∗} : π_{1}(Σ(p, q, r)) → π_{1}(Σ(p, q, r)),
h ↦ h^{−1}, x ↦ x^{−1}, y ↦ xy^{−1}x^{−1}, z ↦ xyz^{−1}y^{−1}x^{−1},

see [2], Proposition 12.30, which in turn induces an involution on the corresponding representation space ( [·, ·] stands for conjugacy class),

σ^{∗} : R(Σ(p, q, r)) → R(Σ(p, q, r)), [α] ↦ [α′],  (8)
where α′(t) = α(σ_{∗}(t)), t ∈ π_{1}(Σ(p, q, r)).  (9)

Lemma 8. If α′ : π_{1}(Σ(p, q, r)) → SU(2) is a representation defined by the formula (9) then there exists an element ρ ∈ SU(2) such that ρ^{2} = −1 and

α′(t) = ρ · α(t) · ρ^{−1} for all t ∈ π_{1}(Σ(p, q, r)).

The element ρ is defined uniquely up to multiplication by ±1, and the elements ρ corresponding to different representations α are conjugate to each other. In particular, the action (8) on the space R(Σ(p, q, r)) of the conjugacy classes of irreducible representations of π_{1}(Σ(p, q, r)) in SU(2) is trivial.

Proof. After conjugation we may assume that α(x) is a unit complex number. Then α′(x) = α(σ_{∗}(x)) = α(x^{−1}) = α(x)^{−1} is a complex number as well. Let us consider another representation, β, defined as a conjugate of α by the unit quaternion j,

β(t) = j^{−1} · α′(t) · j for all t ∈ π_{1}(Σ(p, q, r)).

Next we want to verify that the representations α and β satisfy the conditions of Lemma 6. We have

β(x) = j^{−1} · α′(x) · j
= j^{−1} · α(x)^{−1} · j
= j^{−1} · \overline{α(x)} · j
= j^{−1} · j · α(x), since α(x) ∈ ℂ,
= α(x).

Note that, for any unit quaternion a, we have r(a) = r(ā) = r(a^{−1}). Therefore,

r(β(y)) = r(j^{−1} · α′(y) · j) = r(α′(y))
= r(α(xy^{−1}x^{−1})) = r(α(x)α(y)^{−1}α(x)^{−1})
= r(α(y)^{−1}) = r(α(y)),

and similarly,

r(β(x)β(y)) = r(j^{−1} · α′(x)α′(y) · j) = r(α′(x)α′(y))
= r(α(x)^{−1}α(x)α(y)^{−1}α(x)^{−1}) = r((α(x)α(y))^{−1})
= r(α(x)α(y)).

Since r(α(h)) is equal to r(β(h)) we can apply Lemma 6 to the representations α and β to find a complex number c ∈ ℂ such that β(t) = c · α(t) · c^{−1} for all t ∈ π_{1}(Σ(p, q, r)). Since β(t) = j^{−1}·α′(t)·j = c·α(t)·c^{−1}, we get that α′(t) = ρ·α(t)·ρ^{−1} with ρ = jc.

If we apply σ^{∗} twice to the representation α, we will get α again because σ^{∗} is an involution. Therefore, α is conjugate to itself by the element ρ^{2}. But the representation α is irreducible, therefore, ρ^{2} = ±1. The following easy computation shows that ρ^{2} is in fact −1,

ρ^{2} = jcjc = jjc̄c = −|c|^{2} = −1.

The uniqueness of ρ up to ±1 follows from the irreducibility of α. The fact that the elements ρ defined by different representations are conjugate follows from the fact that tr(jc) = 0 for any complex number c.

4. The definition of λ^{ρ}. Let Σ = Σ(p, q, r) be a Brieskorn homology sphere with a σ–invariant Heegaard splitting

Σ = M_{1} ∪_{M_{0}} M_{2}  (10)

constructed in (6). According to Lemma 8, for any representation α : π_{1}(Σ(p, q, r)) → SU(2), there exists an element ρ ∈ SU(2) such that α ∘ σ_{∗} = ραρ^{−1} and ρ^{2} = −1. Generally speaking, ρ depends on α but it follows from Lemma 8 that the elements ρ corresponding to different α’s are conjugate to each other. Therefore, the following definition makes sense.

For each manifold M_{i}, i = 0, 1, 2, in (10) we define the so called ρ–invariant representation space,

R^{ρ}(M_{i}) = { α : π_{1}(M_{i}) → SU(2) | α is irreducible and

α(σ_{∗}(t)) = ρα(t)ρ^{−1}, t ∈ π_{1}(M_{i}) }/ ad SU(2)^{ρ}, (11)

as the space of all irreducible ρ–invariant representations modulo the adjoint action of the 1–dimensional Lie group SU(2)^{ρ} ⊂ SU(2) consisting of all g ∈ SU(2) such that ρg = ±gρ. Note that whenever ρ′ = ±hρh^{−1}, h ∈ SU(2), we have that R^{ρ}(M_{i}) = R^{ρ′}(M_{i}), hence each of the spaces (11) only depends on the conjugacy class of ρ and not on ρ itself. We do not need to define a ρ–invariant representation space R^{ρ}(Σ(p, q, r)) because all the representations of π_{1}(Σ(p, q, r)) in SU(2) are ρ–invariant, see Lemma 8, and therefore R^{ρ}(Σ(p, q, r)) = R(Σ(p, q, r)).

Due to van Kampen’s theorem, we have the following commutative diagrams of the fundamental groups

π_{1}(Σ(p, q, r)) ⟵ π_{1}(M_{1})
↑  ↑
π_{1}(M_{2}) ⟵ π_{1}(M_{0})

and of the ρ–invariant representation spaces

R(Σ(p, q, r)) ⟶ R^{ρ}(M_{1})
↓  ↓
R^{ρ}(M_{2}) ⟶ R^{ρ}(M_{0})

The maps in the latter diagram are injective, so we can think about R^{ρ}(Σ(p, q, r)) as the intersection of R^{ρ}(M_{1}) and R^{ρ}(M_{2}) inside R^{ρ}(M_{0}).

Proposition 9. The ρ–invariant representation varieties R^{ρ}(M_{i}), i = 0, 1, 2, are smooth manifolds of dimensions R^{ρ}(M_{i}) = 2g −1, i = 1, 2, and dim R^{ρ}(M_{0}) = 4g −2 where g is the genus of a σ–invariant Heegaard splitting (10). The maps in the second diagram are embeddings. The submanifolds R^{ρ}(M_{1}) and R^{ρ}(M_{2}) of R^{ρ}(M_{0}) intersect transversally, and their intersection is in one-to-one correspondence with R(Σ(p, q, r)).

The manifolds R^{ρ}(M_{i}), i = 0, 1, 2, can be oriented as follows. We start with R^{ρ}(M_{0}). Since the quotient M_{0}/σ is a 2–sphere and the branching set consists of 2n points, n = g + 1, one can choose generators a_{1}, b_{1}, …, a_{g}, b_{g} in π_{1}(M_{0}) such that σ_{∗}(a_{i}) = a_{i}^{−1}, σ_{∗}(b_{i}) = b_{i}^{−1}. Then R^{ρ}(M_{0}) ⊂ Hom^{ρ}(F_{2g}, SU(2))/ ad SU(2)^{ρ} where F_{2g} is a free group on the generators a_{1}, b_{1}, …, a_{g}, b_{g}, and

Hom^{ρ}(F_{2g}, SU(2)) = { α : F_{2g} → SU(2) |

α(a_{i})^{−1} = ρα(a_{i})ρ^{−1}, α(b_{i})^{−1} = ρα(b_{i})ρ^{−1}, i = 1, …, g }.

Lemma 10. Let ρ ∈ SU(2) be such that ρ^{2} = −1. Then the subset S^{ρ} of SU(2) consisting of all a ∈ SU(2) such that a^{−1} = ρaρ^{−1} is a two-dimensional sphere S^{2} ⊂ SU(2).

Proof. Any element ρ of SU(2) with ρ^{2} = −1 has zero trace. Therefore, there exists x ∈ SU(2) such that ρ = xjx^{−1} (remember that we identify SU(2) with the group of unit quaternions), and then the map a ↦ x^{−1}ax establishes a diffeomorphism between S^{j} and S^{ρ}. Now, S^{j} consists of all b ∈ SU(2) such that b̄j = jb and |b|^{2} = 1, hence b = u + vi + wk for some real u, v, w with u^{2} + v^{2} + w^{2} = 1. This of course determines a 2–sphere.

Our choice of a_{1}, b_{1}, …, a_{g}, b_{g} establishes a diffeomorphism

Hom^{ρ}(F_{2g}, SU(2)) = (S^{2})^{2g}.

By choosing an orientation on S^{2}, we orient Hom^{ρ}(F_{2g}, SU(2)) as a product. This also orients R(M_{0}) by the usual “base–fiber” convention. Note that the orientation on R^{ρ}(M_{0}) is independent of our choices of orientations on S^{2} and SU(2)^{ρ} or our choice of a basis a_{1}, b_{1}, …, a_{g}, b_{g}.

The orientation on R^{ρ}(M_{1}) can be defined as follows. First, we choose a basis x_{1}, …, x_{g} of the free group π_{1}(M_{1}) so that σ_{∗}(x_{i}) = x_{i}^{−1}, i = 1, …, g. Then R^{ρ}(M_{1}) ⊂ Hom^{ρ}(F_{g}, SU(2))/ ad SU(2)^{ρ}, where F_{g} is a free group on the generators x_{1}, …, x_{g}, and

Hom^{ρ}(F_{g}, SU(2)) = { α : F_{2g} → SU(2) | α(σ_{∗}(x)) = ρα(x)ρ^{−1}}

= {α : F_{g} → SU(2) | α(x_{i})^{−1} = ρα(x_{i})ρ^{−1}, i = 1, …, g }.

Our choice of x_{1}, …, x_{n} establishes a diffeomorphism

Hom^{ρ}(F_{g}, SU(2)) = (S^{2})^{g}.

We choose an orientation on S^{2} and orient Hom^{ρ}(F_{g}, SU(2)) as a product. Note that this orientation changes by (−1)^{g} when the orientation on S^{2} is changed; however, it does not depend on the choice of a basis in π_{1}(M_{1}) = F_{g}. The orientation of Hom^{ρ}(F_{g}, SU(2)) and an orientation of SU(2)^{ρ} orient R(M_{1}).

The ρ–invariant representation space R^{ρ}(M_{2}) is oriented in a completely similar way, the orientation of SU(2)^{ρ} having already been fixed.

If the orientations on S^{2} or SU(2)^{ρ} are changed, the spaces R(M_{1}) and R(M_{2}) change their orientations simultaneously; therefore, their algebraic intersection number in R^{ρ}(M_{0}) does not change (though a priori it may depend on the chosen Heegaard decomposition). We define our λ^{ρ}–invariant as one half of this algebraic intersection number, so that

λ^{ρ}(Σ(p, q, r)) = 1/2 · ∑_{α∈R(Σ)} ε_{α}  (12)

where ε_{α} equals ±1 depending on whether the orientations on the tangent spaces T_{α}R^{ρ}(M_{1}) ⊕ T_{α}R^{ρ}(M_{2}) and T_{α}R^{ρ}(M_{0}) agree.

Proposition 11. For any Brieskorn homology sphere Σ(p, q, r) the invariant λ^{ρ}(Σ(p, q, r)) is well-defined, in particular, it does not depend on the choice of a σ–invariant Heegaard splitting. Moreover,

λ^{ρ}(Σ(p, q, r)) = 1/8 · sign k(p, q, r)

where sign stands for the knot signature.

This proposition is proven below by pushing invariant representations down to a knot k(p, q, r) complement in S^{3} and applying a result from [8] on trace-free SU(2)– representations of knot groups.

Thus our definition of λ^{ρ}–invariant is modelled on the Casson’s definition of the λ– invariant; the difference is that we use ρ–invariant representation spaces. Surprisingly enough, we get an invariant which is different from λ. For example, λ(Σ(2, 3, 5)) = λ(Σ(2, 3, 7)) = 1 while λ^{ρ}(Σ(2, 3, 5)) = −1 and λ^{ρ}(Σ(2, 3, 7)) = 1.

5. Casson-Lin invariant. Let B_{n} be the braid group of rank n with the standard generators β_{1}, …, β_{n−1} represented in a free group F_{n} on symbols x_{1}, …, x_{n} as follows:

β_{i} : x_{i} ↦ x_{i}x_{i+1}x_{i}^{−1}
x_{i+1} ↦ x_{i}
x_{j} ↦ x_{j}, if j ≠ i, i + 1.

If β ∈ B_{n} then the automorphism of F_{n} representing β maps each x_{i} to a conjugate of some x_{j} and preserves the product x_{1} ⋯ x_{n}.

Let k ⊂ S^{3} be a knot represented as the closure of a braid β ∈ B_{n}. Let us fix an embedding of k into S^{3} as shown in Figure 2, the sphere S separating S^{3} in two 3–balls, B_{1} and B_{2}, with the braid β inside B_{1} and n untangled arcs inside B_{2}. The fundamental group π_{1}(K) of the knot k complement K = S^{3} − nbd (k), has the presentation

π_{1}(K) = ⟨ x_{1}, …, x_{n} | x_{i} = β(x_{i}), i = 1, …, n ⟩,

the generators x_{1}, …, x_{n} being represented by the meridians of β.

The knot complement K can be now decomposed as K = M′_{1} ∪_{M′_{0}} M′_{2} where M′_{0} = S ∩ K and M′_{i} = B_{i} ∩ K, i = 1, 2. The manifolds M′_{1} and M′_{2} are handlebodies of genus n, and M′_{0} is a 2–sphere with 2n small 2–discs removed around the points P_{1}, …, P_{2n}, see Figure 2. Due to van Kampen’s theorem, we get the following commutative diagram of fundamental groups

π_{1}(K) ⟵ π_{1}(M′_{1})
↑  ↑
π_{1}(M′_{2}) ⟵ π_{1}(M′_{0})

where

π_{1}(M′_{0}) = ⟨ x_{1}, …, x_{n}, y_{1}, …, y_{n} | x_{1} ⋯ x_{n} = y_{1} ⋯ y_{n} ⟩,
π_{1}(M′_{1}) = ⟨ x_{1}, …, x_{n} | ⟩, π_{1}(M′_{2}) = ⟨ y_{1}, …, y_{n} | ⟩

are isomorphic to free groups. To continue an analogy with the definition of the Casson invariant, we need to define SU(2)–representation spaces of the groups π_{1}(K) and π_{1}(M_{i}), i = 0, 1, 2, and compute the corresponding intersection number. To make things work, we impose the extra condition on the representations that all the meridians go to trace-free matrices in SU(2). Thus, following [8], we define

Ĥ_{n} = { α : π_{1}(M′_{0}) → SU(2) | α is irreducible, tr x_{i} = tr y_{i} = 0 }/ ad SU(2),
Γ̂_{β} = { α : π_{1}(M′_{1}) → SU(2) | α is irreducible, tr x_{i} = 0 }/ ad SU(2),
Λ̂_{n} = { α : π_{1}(M′_{2}) → SU(2) | α is irreducible, tr y_{i} = 0 }/ ad SU(2),
R^{0}(K) = { α : π_{1}(K) → SU(2) | α is irreducible, tr x_{i} = 0 }/ ad SU(2).

The first three are smooth manifolds of dimensions dim Ĥ_{n} = 4n − 6, and dim Γ̂_{β} = dim Λ̂_{n} = 2n − 3. The commutative diagram of fundamental groups induces the following commutative diagram

R^{0}(K) ⟶ Γ̂_{β}
↓  ↓
Λ̂_{n} ⟶ Ĥ_{n}

In particular, the irreducible trace-free representations of π_{1}(K) in SU(2) are in oneto-one correspondence with the intersection points of Λ̂_{n} with Γ̂_{β} in Ĥ_{n}. The manifolds Λ̂_{n}, Γ̂_{β}, and Ĥ_{n} are naturally oriented, see [8], and (possibly after a perturbation to make the intersection transversal) one can define

h(k) = ∑_{α∈R^{0}(K)} ε′_{α}  (13)

where ε′_{α} = ±1 is a sign obtained by comparing the orientations on T_{α}Λ̂_{n} ⊕ T_{α}Γ̂_{β} and T_{α}Ĥ_{n}. The invariant h(k) only depends on the knot k and not on the choices in the definition. X.-S. Lin in [8] further proves that h(k) = 1/2 · sign k.

6. Representations of a knot k(p, q, r) complement. Throughout this subsection Σ = Σ(p, q, r) is a Brieskorn homology sphere, and k = k(p, q, r) is the corresponding Montesinos knot.

Let E → Σ be a (necessarily trivial) SU(2)–vector bundle over Σ. Let us fix a Riemannian metric on Σ and consider the Banach manifold

B^{∗} = A^{∗}/G

of the L_{2}^{2}–gauge equivalence classes of irreducible L_{1}^{2}–connections in E. It is a classical result in differential geometry that the holonomy map establishes a one-to-one correspondence between the gauge equivalence classes of irreducible flat connections and the conjugacy classes of irreducible SU(2)–representations of π_{1}(Σ).

The involution σ can be lifted to a bundle endomorphism of E. Any endomorphism of E clearly induces an action on A^{∗} by pull-back, and an action on B^{∗} as well. Since any two liftings of σ differ by a gauge transformation, we have a well-defined action σ^{∗} : B^{∗} → B^{∗}. Denote by B^{σ} the space of connections invariant with respect to σ^{∗}.

Let ρ ∈ SU(2) be such that ρ^{2} = −1. The formula

(x, ξ) ↦ (σ(x), ρξρ^{−1})  (14)

defines a lifting of σ : Σ → Σ on E, which will again be denoted by ρ : E → E. Let B^{ρ} ⊂ B^{σ} consist of the gauge equivalence classes of irreducible connections A in E such that ρ^{∗}A = A. Due to Lemma 8, all irreducible flat connections on Σ belong to B^{ρ}; in particular, B^{ρ} is non-empty. The following three lemmas are easy corollaries of Propositions 1 and 17 and Theorem 18 in [20].

Lemma 12. For any ρ, ρ′ ∈ SU(2) such that ρ^{2} = ρ′^{2} = −1, B^{ρ} = B^{ρ′}. Furthermore, as a set, B^{ρ} is bijective to A^{ρ}/\overline{G}^{ρ} where \overline{G} = G/±1 and A^{ρ} = \{ A ∈ A^{∗} | ρ^{∗}A = A }.

The quotient of Σ(p, q, r) by σ is S^{3}. Since ρ ≠ ±1, it is impossible to define the quotient bundle of E over S^{3}. One can though define it away from the image of the fixed point set of σ, that is, on the knot complement K = S^{3} \ k. Given an irreducible flat ρ–invariant SU(2)–connection A ∈ A^{ρ}, its push-down A′ is an irreducible flat SO(3)–connection on K. In other words, A′ is an irreducible flat SO(3)–connection singular along k in the sense of P. Kronheimer and T. Mrowka, see [7].

Lemma 13. The SO(3)–connection A′ has holonomy 1/2 around k.

According to H^{1}(K; ℤ/2) = ℤ/2 and H^{2}(K; ℤ/2) = 0, there are two different ways to lift the SO(3)–connection A′ to a (singular) SU(2)–connection. Both SU(2)–liftings have holonomy 1/4 around k; in other words, their holonomy around k is trace–free.

Lemma 14. Through the push-down map, the representation space R(Σ) is in oneto-two correspondence with the representation space R^{0}(K) of irreducible trace–free SU(2)–representations of π_{1}(K).

Similar one-to-two identifications through push-down hold for the pairs R^{ρ}(M_{0}) and Ĥ_{n}, R^{ρ}(M_{1}) and Γ̂_{β}, R^{ρ}(M_{2}) and Λ̂_{n}, where g = n − 1 is the genus of the σ–invariant Heegaard splitting, see Figure 2.

Proof of Proposition 11. Remember that throughout this subsection, Σ = Σ(p, q, r) and k = k(p, q, r). The invariants λ^{ρ}(Σ) and h(k) = 1/2 · sign k were defined in (12) and (13) as an algebraic count of points in the corresponding (finite) representation spaces, R(Σ) and R^{0}(K). These spaces are in one-to-two correspondence by Lemma 14. A routine comparison of the orientations shows that ε_{α} = ε′_{α′} where α′ is a push-down of α. Therefore,

λ^{ρ}(Σ) = 1/2 · ∑ ε_{α}
= 1/4 · ∑ ε′_{α′}
= 1/8 · sign k.

This completes the proof of Proposition 11.

3. Gauge theory for Brieskorn homology spheres

Our next step is to give an “analytic” definition of the invariant λ^{ρ}. In [19], C. Taubes gave a gauge-theoretical interpretation of the Casson’s λ–invariant in terms of (what later became known as) Floer homology. Our gauge-theoretical interpretation of λ^{ρ} will follow the same lines with the only difference that everything will be “ρ–invariant”.

1. Casson invariant via gauge theory. Let us recall shortly the Taubes construction [19] for a homology 3–sphere Σ. Let E → Σ be a trivial SU(2)–bundle, and B^{∗} = A^{∗}/G the space of gauge equaivalence classes of irreducible connections in E. It is a classical differential geometry result that flat connections in E (modulo gauge equivalence) are in one-to-one correspondence with SU(2)–representations of π_{1}(Σ) (modulo conjugation). Given two irreducible representations α, β, we may think of them as points in A^{∗} or B^{∗}.

For any A ∈ A^{∗}, define the following elliptic differential operator

K_{A} = |\begin{matrix}0 & d_{A}^{∗} \\ d_{A} & ∗d_{A} \end{matrix}| : (Ω^{0} ⊕ Ω^{1})(Σ, 𝔰𝔲(2)) → (Ω^{0} ⊕ Ω^{1})(Σ, 𝔰𝔲(2)),  (15)

where d_{A} stands for the operator of covariant differentiation with respect to A, and ∗ is the Hodge operator associated with a Riemannian metric on Σ. In a proper Sobolev completion of (Ω^{0} ⊕Ω^{1})(Σ, 𝔰𝔲(2)), the operator K_{A} is Fredholm. Let A(t), 0 ≤ t ≤ 1, be a path in A^{∗} connecting α with β, so that A(0) = α, A(1) = β. Associated with A(t) is a path of Fredholm operators, K_{A(t)}. Let us assume that ker K_{A(0)} = ker K_{A(1)}, as is the case for a Brieskorn homology sphere Σ = Σ(p, q, r). We define sf(α, β) as the spectral flow of the path K_{A(t)} between α and β, see [1]. This number is welldefined modulo 8, and Taubes defines an infinite dimensional generalization χ(Σ) of the Euler characteristic as

χ(Σ) = ε_{α} · ∑_{β∈R(Σ)}(−1)^{sf(α,β)}  (16)

where ε_{α} is figured out from the spectral flow between the trivial connection θ and α (the number χ(Σ) turns out to be independent of α). It is proven in [19] that λ(Σ) = 1/2 · χ(Σ). Later, A. Floer showed in [5] that there are well–defined ℤ/8– graded instanton homology groups I_{∗}(Σ) such that χ(Σ) = χ(I_{∗}(Σ)).

Let now Σ = Σ(p, q, r) be a Brieskorn homology sphere. R. Fintushel and R. Stern showed in [6] (see also Proposition 15 below) that for any pair α, β ∈ R(Σ(p, q, r)),

sf(α, β) = 0 mod 2,

and that all the signs involved in computing χ(Σ(p, q, r)) are positive, so χ simply counts the irreducible representations.

2. Floer index for Brieskorn homology spheres. Let α and β be irreducible flat connections in a trivial SU(2)–bundle E over Σ = Σ(p, q, r). By pull-back, we can extend E to a trivial bundle (which we also denote by E) over the infinite cylinder Σ × ℝ. Let us choose a σ–invariant Riemannian metric on Σ, and the corresponding product metric on Σ × ℝ. Let further ρ : E → E be the lifting of σ defined in (14), and A(t) ∈ A^{ρ} a path of connections forming an invariant connection A over Σ × ℝ vanishing in the ℝ–direction and equal to respectively α and β near the ends of Σ×ℝ. Denote by d_{A} the operator of covariant differentiation with respect to A.

The (relative) Floer index of the pair (α, β) equals the Fredholm index of the following elliptic complex, see [1],

Ω^{0}(Σ × ℝ, ad E) ⟶^{d_{A}} Ω^{1}(Σ × ℝ, ad E) ⟶^{d_{A}^{−}} Ω_{+}^{2}(Σ × ℝ, ad E)  (17)

where ad E is the adjoint bundle of E over Σ × ℝ, and d_{A}^{−} = P_{−} ∘ d_{A} where P_{−} is the projection onto the self dual forms with respect to the fixed product metric on Σ × ℝ. The Sobolev norms on the spaces in (17) are specified as usual, see [5].

The product metric and the connection A on Σ × ℝ are ρ–invariant, therefore, one can define the following ρ–invariant elliptic subcomplex of (17),

Ω^{0}(Σ × ℝ, ad E)^{ρ} ⟶^{d_{A}} Ω^{1}(Σ × ℝ, ad E)^{ρ} ⟶^{d_{A}^{−}} Ω_{+}^{2}(Σ × ℝ, ad E)^{ρ}  (18)

Proposition 15. For any irreducible flat connections α, β on Σ = Σ(p, q, r), the index of the elliptic complex (17) is even. In fact, it equals twice the index of the ρ–invariant complex (18).

In order to prove the proposition we need the following two technical lemmas.

Lemma 16. There exist a Riemannian metric on Σ(p, q, r) and an almost complex structure J on Σ(p, q, r) × ℝ such that

(1) The product metric on Σ × ℝ is (σ × 1)–invariant;

(2) J is compatible with this product metric; and

(3) The involution σ × 1 is anti-holomorphic with respect to J, that is (σ × 1)_{∗}J = −J(σ × 1)_{∗} on the tangent bundle.

Proof. Let us consider the algebraic variety

V (p, q, r) = { (x, y, z) ∈ ℂ^{3} | x^{p} + y^{q} + z^{r} = 0 } ⊂ ℂ^{3},

which is a non-singular complex surface except perhaps at the origin. The homology sphere Σ(p, q, r) in question is the intersection of V (p, q, r) with the unit 5-sphere S_{1}^{5},

Σ(p, q, r) = V (p, q, r) ∩ S_{1}^{5},

and the variety V (p, q, r) is in fact a cone over Σ(p, q, r) with the vertex at the origin.

Let V^{0}(p, q, r) be the variety V (p, q, r) with the origin removed. The Riemann metric induced on V^{0}(p, q, r) from the standard flat metric on ℂ^{3} is a cone metric given by

ds^{2} = dr^{2} + r^{2}dθ^{2}  (19)

in the spherical coordinates (r, θ). Here dθ^{2} = ∑ h_{ij}dθ_{i}dθ_{j} is a metric on Σ(p, q, r), and r is the distance from the origin (in ℂ^{3}). Obviously, both the complex conjugation σ′ and the almost complex structure J induced on V^{0}(p, q, r) from ℂ^{3} are compatible with the metric (19).

Let us now form the conformally equivalent metric

ds^{2}/r^{2} = dr^{2}/r^{2} + dθ^{2}  (20)

on V^{0}(p, q, r). The substitution r = e^{−τ} gives coordinates in which ds^{2}/r^{2} is the standard product metric

ds̃^{2} = dτ^{2} + dθ^{2}  (21)

on the cylinder Σ(p, q, r) × ℝ. Since the metrics (19) and (21) are conformally equivalent, both σ′ and J are compatible with the product metric (21). Moreover, σ′ preserves all the spheres S_{r}^{5}, 0 < r < ∞, in ℂ^{3}, therefore, it has the form σ′ = σ × 1. Finally, the involution σ × 1 is antiholomorphic with respect to J because this is the case in ℂ^{3}.

Lemma 17. There exists a differential operator d̃_{A}^{−} : Ω^{1}(Σ × ℝ, ad E) → Ω_{+}^{2}(Σ × ℝ, ad E) such that

(1) The difference d̃_{A}^{−} − d_{A}^{−} is a compact operator,

(2) coker d̃_{A}^{−} = 0, and

(3) The operator d̃_{A}^{−} is ρ–invariant.

Proof. This is essentially the transversality result from [5], Proposition 2c.2. The key difference is that we have to bound ourselves to invariant perturbations which, generally speaking, may be not generic among the perturbations used by Floer. Following [5], Section 1b, consider a collection S(m) of m circles smoothly embedded in ℝ^{3} which intersect precisely at the origin and have the same tangent there. Let

γ : ⋁_{i=1}^{m} S_{i}^{1} × D^{2} → Σ,

be a map which restricts to smooth embeddings γ_{θ} : S(m) → Σ for each θ ∈ D^{2} and γ_{i} : S_{i}^{1} × D^{2} → Σ for each i. By adding extra circles, one can choose γ so that its image in Σ will be σ–invariant. Now, γ defines a family of holonomy maps

γ_{θ} : B → L_{m} = SU(2)^{m}/ ad SU(2),
A ↦ (hol_{A}(γ_{1}), …, hol_{A}(γ_{m}))  (22)

Let C_{sym}^{∞}(L_{m}, ℝ) be defined as the set of smooth ad SU(2)–invariant real valued functions on SU(2)^{m} with the additional property that for any h ∈ C_{sym}^{∞}(L_{m}, ℝ),

h(x_{τ(1)}, …, x_{τ(m)}) = h(x_{1}, …, x_{m})

for any permutation τ on m symbols. For example, one can start with an arbitrary ad SU(2)–invariant function on SU(2)^{m} and take h to be its symmetrization.

Given such γ and h, we define the function

h_{γ} : B → ℝ, h_{γ}(A) = ∫_{D^{2}} h(γ_{θ}(A)) d^{2}θ  (23)

where d^{2}θ is a smooth compactly supported volume form on the interior of D^{2}. The following argument shows that this function is invariant with respect to the induced σ^{∗}–action on B:

h_{γ}(σ^{∗}A) = ∫_{D^{2}} h( hol_{σ^{∗}A}(γ_{1}), …, hol_{σ^{∗}A}(γ_{m})) d^{2}θ

= ∫_{D^{2}} h( hol_{A}(σ(γ_{1})), …, hol_{A}(σ(γ_{m}))) d^{2}θ
= ∫_{D^{2}} h( hol_{A}(γ_{τ(1)}), …, hol_{A}(γ_{τ(m)})) d^{2}θ

= ∫_{D^{2}} h( hol_{A}(γ_{1}), …, hol_{A}(γ_{m})) d^{2}θ = h_{γ}(A).

The class of maps we defined in (23) is large enough for the set

{ grad h_{γ}(A) | h_{γ} defined by (23) } ⊂ T_{A}B^{∗}

to be dense in the tangent space T_{A}B^{∗}. Now the proof of Proposition 2c.2 in [5] goes through, which provides a function h_{γ} such that a linearization of its gradient is a compact invariant perturbation of the operator d_{A}^{−} with the desired properties.

Sketch of the proof of Proposition 15. In Lemma 16 we defined a σ–invariant Riemannian metric on Σ(p, q, r) and an almost complex structure J on the manifold Σ(p, q, r) × ℝ compatible with the product metric such that the involution σ × 1 : Σ × ℝ → Σ × ℝ is anti-holomorphic with respect to J. We can use J to identify the ±1–eigenspaces of the involution induced by σ×1 on the space of 1–forms in (17), and make sure that (possibly after perturbation provided by Lemma 17) ker d_{A} = 0 and coker d_{A}^{−} = 0. This splits the index of the complex (17) and hence the corresponding spectral flow in halves.

Proposition 18. ( Compare [6] ). Let Σ(p, q, r) be a Brieskorn homology sphere. Then the Floer homology groups I_{n}(Σ(p, q, r)) vanish for n odd.

Proof. We only need to fix the sign ε_{α} in (16). It can be achieved by the requirement that for ε_{α} = (−1)^{η(α)},

−3 − η(α) = index D_{A}^{+} − 3 (1 + b_{2}^{+})(W),

where W is any smooth simply connected non-compact 4–manifold with a single end of the form Σ(p, q, r) × ℝ_{+}, A is any SU(2)–connection on W with limiting value α, and

D_{A}^{+} = d_{A} ⊕ d_{A}^{+} : Ω^{1}(W, 𝔰𝔲(2)) → Ω^{0}(W, 𝔰𝔲(2)) ⊕ Ω_{+}^{2}(W, 𝔰𝔲(2))

is the ASD–operator, compare [4]. Let W be the Milnor fiber of the singularity of f^{−1}(0) where f(x, y, z) = x^{p} +y^{q} +z^{r} so that ∂W = Σ(p, q, r), see [9]. The involution (5) extends to an antiholomorphic involution on W, and an argument similar to that in the proof of Proposition 15 proves that index D_{A}^{+} is even. On the other hand, one can easily show that modulo 2, b_{2}^{+} = 1/2 · b_{2}(W) = 1/2 · (p − 1)(q − 1)(r − 1) = 0. Therefore, ε_{α} = 1, and the Floer homology groups of Σ(p, q, r) vanish in odd dimensions.

3. The definition of χ^{ρ}. Let ρ : E → E be the lifting of σ : Σ(p, q, r) → Σ(p, q, r) defined by (14). It induces an involution ρ^{∗} on the 𝔰𝔲(2)–valued differential forms on Σ(p, q, r). Therefore, for any A ∈ A^{ρ}, the differential operator K_{A}, see (15), can be restricted to the +1–eigenspaces of the involution ρ^{∗} to give a new elliptic differential operator K_{A}^{ρ},

K_{A}^{ρ} = |\begin{matrix}0 & d_{A}^{∗} \\ d_{A} & ∗d_{A} \end{matrix}| : (Ω^{0} ⊕ Ω^{1})^{ρ}(Σ, 𝔰𝔲(2)) → (Ω^{0} ⊕ Ω^{1})^{ρ}(Σ, 𝔰𝔲(2)).

For any pair α, β ∈ R(Σ(p, q, r)), let us denote by sf^{ρ}(α, β) the spectral flow of a family of operators K_{A(t)}^{ρ}, where A(t) is a path in A^{ρ} connecting α to β. The number sf^{ρ}(α, β) is well-defined modulo 4. The invariant χ^{ρ}(Σ(p, q, r)) is now defined by the formula

χ^{ρ}(Σ) = ε_{α} · ∑_{β∈R(Σ)}(−1)^{sf^{ρ}(α,β)},  (24)

compare to (16). The following result is an easy consequence of Proposition 15.

Proposition 19. Let Σ = Σ(p, q, r) be a Brieskorn homology sphere. Then 1/2 · χ^{ρ}(Σ) = ν(Σ) where the invariants χ^{ρ}(Σ) and ν(Σ) are defined by (24) and (1).

4. The invariant λ^{ρ} via gauge theory

Proposition 20. Let Σ = Σ(p, q, r) be a Brieskorn homology sphere, and λ^{ρ} and χ^{ρ} the invariant defined by (12) and (24). Then λ^{ρ}(Σ) = 1/2 · χ^{ρ}(Σ).

This result is a straightforward application of the Taubes argument in [19] in our ρ–invariant setting.

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

  E-mail address: saveliev@math.lsa.umich.edu