SEIBERG-WITTEN THEORY AND Z/2^{p} ACTIONS ON SPIN 4-MANIFOLDS

JIM BRYAN

Abstract. Furuta’s “10/8-th’s” theorem gives a bound on the magnitude of the signature of a smooth spin 4-manifold in terms of the second Betti number. We show that in the presence of a Z/2^{p} action, his bound can be strengthened. As applications, we give new genus bounds on classes with divisibility and we give a classification of involutions on rational cohomology K3’s.

We utilize the action of P in(2) ˜×Z/2^{p} on the Seiberg-Witten moduli space. Our techniques also provide a simplification of the proof of Furuta’s theorem.

1. Introduction

In early 1995, Furuta [6] proved that if X is a smooth, compact, connected spin 4-manifold with non-zero signature σ(X), then

\frac{5}{4}|σ(X)| + 2 ≤ b_{2}(X). (1)

This estimate has been dubbed the “10/8-th’s” theorem in comparison with the “11/8-th’s” conjecture which predicts the following bound:

\frac{11}{8}|σ(X)| ≤ b_{2}(X).

The inequality (1) follows by a surgery argument from the non-positive signature, b_{1}(X) = 0 case:

Theorem 1.1 (Furuta). Let X be a smooth spin 4-manifold with b_{1}(X) = 0 with non-positive signature. Let k = −σ(X)/16 and m = b_{2}^{+}(X). Then,

2k + 1 ≤ m

if m ≠ 0.

The main result of this paper improves the above bound by p under the assumption that X has a Z/2^{p} action satisfying some non-degeneracy conditions (the analogues of the condition m ≠ 0 in the above theorem).

A Z/2^{p} action is called a spin action if the generator of the action τ : X → X lifts to the spin bundle τ̂ : P_{Spin} → P_{Spin}. Such an action is of even type if τ̂ has order 2^{p} and is of odd type if τ̂ has order 2^{p+1}. Using [2], it easy to determine if an action is spin and whether it is of even or odd type.

Supported by M.S.R.I. postdoctoral fellowship.

We assume throughout that X is a smooth spin 4-manifold with b_{1}(X) = 0 and oriented so that the signature is non-positive. We will continue with the notation k = −σ(X)/16 and m = b_{2}^{+}(X). Our main results are the following:

Theorem 1.2. Suppose that τ : X → X generates a smooth Z/2^{p} action that is spin and of odd type. Let X_{i} denote the quotient of X by Z/2^{i} ⊂ Z/2^{p}. Then

2k + 1 + p ≤ m

if m ≠ 2k + b_{2}^{+}(X_{1}) and b_{2}^{+}(X_{i}) ≠ b_{2}^{+}(X_{j}) > 0 for i ≠ j.

For even actions, the easiest results to state are for involutions:

Theorem 1.3. Suppose that σ : X → X is a smooth involution of even type. Then

2k + 2 ≤ m

if m ≠ b_{2}^{+}(X/σ) > 0.

This is a special case of the following theorem:

Theorem 1.4. Suppose that σ_{1}, …, σ_{q} : X → X are smooth involutions of even type generating an action of (Z/2)^{q}. Then

2k + 1 + q ≤ m

if m ≠ b_{2}^{+}(X/g) for any non-trivial element g ∈ (Z/2)^{q} and b_{2}^{+}(X/(Z/2)^{q}) ≠ 0.

Remark 1.1. The non-degeneracy conditions in Theorems 1.2, 1.3, and 1.4 can be restated in terms of the fixed point set although for our applications, the conditions on the quotient are more convenient.

We also include the following

Theorem 1.5. Suppose that τ : X → X generates a smooth Z/2^{p} action that is spin and of either type. Then b_{2}^{+}(X/τ) ≠ 0 if k > 0.

One can apply Theorem 1.2 to a cover of a four manifold branched along a smoothly embedded surface. The inequality can be used to get a bound on the genus of the embedded surface (see Theorem 4.2). One special case is the following:

Theorem 1.6. Let M be a smooth, compact, oriented, simply connected 4-manifold (not necessarily spin) with b_{2}^{+}(M) > 1 and let Σ ↪ M be a smooth embedding of a genus g surface. Suppose that the homology class defined by Σ is divisible by 2 and that [Σ]/2 ≡ w_{2}(M) mod 2. Then

g ≥ \frac{5}{4} (\frac{[Σ]^{2}}{4} − σ(M)) − b_{2}(M) + 2.

Z/2^{p} ACTIONS ON SPIN 4-MANIFOLDS 3

This bound is typically weaker than the adjunction inequality from the Seiberg-Witten invariants (see [9]), but the bound applies to manifolds even with zero Seiberg-Witten invariants. For examples where the bound is sharp and new we have (see also example 4.1):

Corollary 1.7. The minimal genus of an embedded surface in CP^{2}#CP^{2} representing the class (6, 2) is 10. The minimal genus of an embedded surface in S^{2} ×S^{2}#CP^{2} representing the class ((4, 4), 6) is 19.

Proof. In each case the lower bound of Theorem 1.6 is realized by the connected sum of smooth algebraic curves in each factor.

As another application of the theorems we give a “classification” of involutions on rational cohomology K3’s.

Theorem 1.8. Let σ : X → X be a spin involution of a rational cohomology K3 (i.e. b_{1}(X) = 0 and Q_{X} ≅ Q_{K3} ). If σ is of even type then it has exactly 8 isolated fixed points and b_{2}^{+}(X/σ) = 3; if σ is of odd type then b_{2}^{+}(X/σ) = 1.

This theorem recovers as a special case a theorem of Donaldson concerning involutions on the K3 ([5] Cor. 9.1.4) and is related to a theorem of Ruberman [11]. We also remark that both possibilities in the theorem actually occur.

The proof of Theorems 1.2, 1.3, 1.4, and 1.5 uses Furuta’s technique of “finite dimensional approximation” for the Seiberg-Witten moduli spaces to reduce the problem to algebraic topology. The main innovation of our technique is our approach to the equivariant K-theory. In particular, we do not need the Adam’s operations in equivariant K-theory and can thus simplify that part of Furuta’s proof. In section 2 we introduce the equations and use Furuta’s technique to study the moduli space; in section 3 we use equivariant K-theory and representation theory to study the Gequivariant properties of the moduli space and prove the main theorems; section 4 is devoted to applications, primarily genus bounds obtained by branched covers and our classification of involutions on rational cohomology K3’s.

We would like to thank R. Cohen, B. Gompf, C. Gordon, D. Kotschick, P. Kronheimer, T. Lawson, N. C. Leung, G. Matíc, P. Oszvath, D. Ruberman, R. Stern, P. Teichner, and R. Wentworth for helpful conversations, and we would like to thank M.S.R.I. for providing an excellent research environment where this work was carried out.

2. Seiberg-Witten Theory

In this section we use Furuta’s “finite dimensional approximation” technique to study the Seiberg-Witten moduli space in the presence of our Z/2^{p} symmetry. The goal of the section is to use the Seiberg-Witten solutions to produce a certain Gequivariant map between spheres. This map will then be studied by algebraic topology in section 3 to produce a proof of the main theorems.

Let X be a smooth, compact, connected, spin 4-manifold with b_{1}(X) = 0 and fix an orientation so that σ(X) ≤ 0. By Rochlin’s theorem, σ(X) is divisible by 16 and so let k = −σ(X)/16 and let m = b_{2}^{+}(X). Let τ : X → X be an orientation preserving diffeomorphism generating a Z/2^{p} action and fix an invariant Riemannian metric g. We assume that τ is a spin action. By definition this means that τ^{∗}(℘) − ℘ is zero as an element of H^{1}(X; Z/2) where ℘ is the spin structure (the difference of two spin structures is naturally an element of H^{1}(X; Z/2)). There is then a lift τ̂ of τ to the spin bundle:

P_{Spin} ⟶^{τ̂} P_{Spin}
↓  ↓
P_{SO(4)} ⟶^{dτ} P_{SO(4)}
↓  ↓
X ⟶^{τ} X.

The order of τ̂ is either 2^{p} or 2^{p+1} depending on whether (τ̂)^{2^{p}} is the identity or the non-trivial deck transformation of the double cover P_{Spin} → P_{SO(4)}. We say that τ is of even type if τ̂ has order 2^{p} and of odd type if τ̂ has order 2^{p+1}. A lemma of Atiyah and Bott makes it easy to determine the type of τ: The 2^{p−1}-th iteration of τ is an involution σ of X and will have a fixed point set consisting of manifolds of constant even dimension. If the fixed point set of σ consists of points (or all of X), then τ is of even type; if the fixed point set is dimension 2, then τ is of odd type. In the case where σ acts freely, then τ is of even or odd type depending on whether the quotient X/σ is spin or not (c.f. [2] or [4]).

On a spin manifold, the Seiberg-Witten equations for the trivial spin^{C} structure take on a somewhat special form. In [6], Furuta gives a concise and elegant description of the equations. We give a slightly different description to avoid repetition and to make the exposition more closely match standard notation.

Let S = S^{+} ⊕ S^{−} denote the decomposition of the spinor bundle into the positive and negative spinor bundles. Clifford multiplication induces an isomorphism

ρ : Λ_{C}^{∗} → End_{C}(S)

between the bundle of complex valued forms and endomorphisms of S. Let ∂/ : Γ(S^{+}) → Γ(S^{−}) be the Dirac operator. The Seiberg-Witten equations are for a pair (a, ϕ) ∈ Ω^{1}(X, \sqrt{−1}R) × Γ(S^{+}) and they are

∂/ϕ + ρ(a)ϕ = 0,
ρ(d^{+}a) − ϕ ⊗ ϕ^{∗} + \frac{1}{2}|ϕ|^{2}1l = 0,
d^{∗}a = 0.

This system of equations is elliptic as written—the last equation defines a slice for the (based) gauge group of the usual equations. The image of \sqrt{−1}Λ_{+}^{2} under ρ is the tracefree, hermitian endomorphisms of S^{+} which we denote by \sqrt{−1}𝔰𝔲(S^{+}). (We remark that by taking a slightly different gauge fixing condition, one could combine the last two equations in the single equation ρ(d^{+}a + d^{∗}a) = ϕ ⊗ ϕ^{∗}.)

Following Furuta’s notation, we regard the equations as the zero set of a map

D + Q : V → W′

where V is the L_{2}^{4}-completion of Γ(\sqrt{−1}Λ^{1} ⊕ S^{+}) and W′ is the L_{2}^{3} completion of Γ(S^{−} ⊕ \sqrt{−1}𝔰𝔲(S^{+}) ⊕ \sqrt{−1}Λ^{0}) and D and Q are the linear and quadratic pieces of the equations respectively, i.e.

D(a, ϕ) = (∂/ϕ, ρ(d^{+}a), d^{∗}a),
Q(a, ϕ) = (ρ(a)ϕ, ϕ ⊗ ϕ^{∗} − \frac{1}{2}|ϕ|^{2}1l, 0).

The image of D + Q is L^{2}-orthogonal to the constant functions in \sqrt{−1}Ω^{0} ⊂ W′. We define W to be the orthogonal complement of the constant functions in W′ and consider D + Q as defined on W:

D + Q : V → W.

We now wish to determine the group of symmetries of the equations. As in Furuta, we will see that Pin (2) acts and, as one would expect, there are additional symmetries arising from τ. The symmetry group turns out to depend on whether τ is of even or odd type.

Define Pin (2) ⊂ SU(2) to be the centralizer of S^{1} ⊂ SU(2). Regarding SU(2) as the group of unit quaternions and taking S^{1} to be elements of the form e^{\sqrt{−1}θ}, Pin (2) then consists of elements of the form e^{\sqrt{−1}θ} or e^{\sqrt{−1}θ}J (our quaternions are spanned by ⟨1, \sqrt{−1}, J, \sqrt{−1}J⟩). We define the action of Pin (2) on V and W as follows: Since S^{+} and S^{−} are SU(2) bundles, Pin (2) naturally acts on Γ(S^{±}) by multiplication on the left. Z/2 acts on Γ(Λ_{C}^{∗}) by multiplication by ±1 and this pulls back to an action of Pin (2) by the natural map Pin (2) → Z/2. A calculation shows that this pullback also describes the induced action of Pin (2) on \sqrt{−1}𝔰𝔲(S^{+}). Both D and Q are then seen to be Pin (2) equivariant maps. Note that the action of S^{1} ⊂ Pin (2) is the ordinary action of the constant gauge transformations.

The isometry τ acts on V and W by pull back by dτ and τ̂; D and Q are equivariant with respect to this action. If τ is of even type, then it induces an action of Z/2^{p} on V and W; if τ is of odd type then it induces an action of Z/2^{p+1}. In the even case the symmetry group is thus

G_{ev} ≡ Pin (2) × Z/2^{p}.

In the odd case Pin (2)×Z/2^{p+1} acts but we see that the diagonal Z/2 subgroup acts trivially: the 2^{p}-th iteration of τ̂ acts on the spinors by the action induced by the non-trivial deck transformation; this is the same as the action of the constant gauge transformation −1 ⊂ S^{1} ⊂ Pin (2). Thus the symmetry group for the odd case is

G_{odd} ≡ \frac{Pin (2) × Z/2^{p+1}}{Z/2} .

Throughout the sequel we write G to mean G_{odd} or G_{ev}.

To state Furuta’s finite dimensional approximation theorem we define V_{λ} for any λ ∈ R to be the subspace of V spanned by the eigenspaces of D^{∗}D with eigenvalues less than or equal to λ. Similarly, define W_{λ} using DD^{∗} and write p_{λ} for the L^{2}- orthogonal projection from W to W_{λ}. Define

D_{λ} + Q_{λ} : V_{λ} → W_{λ}

by the restriction of D + p_{λ}Q to V_{λ}.

Since the eigenspaces of D^{∗}D and DD^{∗} are G invariant and p_{λ} is a G-map, D_{λ}+Q_{λ} is a G equivariant map between finite dimensional G representations and it is an approximation to D + Q in the following sense:

Lemma 2.1 (Lemma 3.4 of [6]). There exists an R ∈ R such that for all λ > R the inverse image of zero (D_{λ} + Q_{λ})^{−1}(0) is compact.

The lemma allows Furuta to use D_{λ} + Q_{λ} to construct a G-equivariant map on disks preserving boundaries

f_{λ} : (BV_{λ,C}, SV_{λ,C}) → (BW_{λ,C}, SW_{λ,C}).

Here V_{λ,C} = V_{λ} ⊗ C, BV_{λ,C} is homotopic to a ball in V_{λ,C}, and SV_{λ,C} is the boundary of BV_{λ,C} with similar definitions for W_{λ,C}, BW_{λ,C}, and SW_{λ,C}.

The virtual G-representation [V_{λ}] − [W_{λ}] ∈ R(G) is the G-index of D and can be determined by the G index theorem and is independent of λ. We discuss its computation in the next section.

3. Equivariant K-theory

In this section we use equivariant K-theory to deduce restrictions on the map f_{λ} : BV_{λ,C} → BW_{λ,C}. Combining this with the index theorem determining [V_{λ}] − [W_{λ}] ∈ R(G) we will prove the main theorems. Our K-theoretic techniques avoid Furuta’s use of the equivariant Adam’s operations and thus also provide a simplification in the proof of his “10/8-th’s” theorem.

3.1. The representation ring of G_{ev} and G_{odd}. We write R(Γ) for the complex representation ring of a compact Lie group Γ and we write direct sum and tensor product of representations additively and multiplicatively respectively. The representation ring R(Z/2^{p}) of Z/2^{p} is isomorphic to the group ring Z(Z/2^{p}) and is generated by the standard one dimensional representation ζ . We write 1 for the trivial representation, so for example ζ^{2^{p}} = 1, and as a Z module, R(Z/2^{p}) is generated by 1, ζ, …, ζ^{2^{p}−1}.

The group Pin (2) has one non-trivial one dimensional representation which we denote ˜1 given by pulling back the non-trivial Z/2 representation by the map Pin (2) → Z/2. It has a countable series of 2 dimensional irreducible representations h_{1}, h_{2}, . . .. The representation h_{1}, which we sometimes write as h, is the restriction of the standard representation of SU(2) to Pin (2) ⊂ SU(2). The representations h_{i} can be obtained using the relation h_{i}h_{j} = h_{i+j} + h_{|i−j|} where by convention h_{0} denotes 1 + ˜1. Note that ˜1 · h_{i} = h_{i}. Let θ denote the standard 1 dimensional representation of S^{1} so that R(S^{1}) ≅ Z[θ, θ^{−1}]. It is easy to see that h_{i} restricts to θ^{i} + θ^{−i} as an S^{1} representation.

Since G_{ev} = Pin (2) × Z/2^{p}, the representation ring is just the tensor product R(G_{ev}) = R(Pin (2)) ⊗ R(Z/2^{p}). We can thus write a general element β ∈ R(G_{ev}) as

β = β_{0}(ζ)1 + β̃_{0}(ζ)˜1 + ∑_{i=1}^{∞} β_{i}(ζ)h_{i}

where β_{0}, β̃_{0}, β_{1}, … are degree 2^{p} − 1 polynomials in ζ and all but a finite number of the β_{i}’s are 0.

The irreducible representations of G_{odd} = (Pin (2) × Z/2^{p+1})/(Z/2) are the representations of Pin (2) × Z/2^{p+1} that are invariant under Z/2. To avoid confusion we will use ξ for the generator of R(Z/2^{p+1}). The Pin (2) × Z/2^{p+1} representations ξ^{i} and h_{i} are non-trivial restricted to Z/2 if and only if i is odd and the representation ˜1 is trivial restricted to Z/2. The subring R(G_{odd}) ⊂ R(Pin (2) × Z/2^{p+1}) is therefore generated by 1, ˜1, ξ^{2}, and ξ^{i}h_{j} where i ≡ j (mod 2). We write a general element β ∈ R(G_{odd}) as

β = β_{0}(ξ) + β̃_{0}(ξ) + ∑_{i=1}^{∞} β_{i}(ξ)h_{i}

where now β_{0}, β̃_{0}, and β_{2i} are even polynomials of degree 2^{p+1} − 2 and the β_{2i+1}’s are odd polynomials of degree 2^{p+1} − 1. In summary we have:

Theorem 3.1. The ring R(G_{ev}) is generated by 1, ˜1, h_{1}, h_{2}, …, and ζ with the relations ζ^{2^{p}} = 1 and h_{i}h_{j} = h_{i+j} + h_{|i−j|}, where h_{0} = 1 + ˜1.

The ring R(G_{odd}) is generated by 1, ˜1, ξ^{2}, h_{2}, h_{4}, …, and ξh_{1}, ξh_{3}, … with the relations ξ^{2^{p+1}} = 1 and h_{i}h_{j} = h_{i+j} + h_{|i−j|}, where h_{0} = 1 + ˜1.

For G = G_{odd} or G_{ev} the restriction map R(G) → R(S^{1}) ≅ Z[θ, θ^{−1}] is given by ˜1 ↦ 1, ξ ↦ 1 (or ζ ↦ 1), and h_{i} ↦ θ^{i} + θ^{−i}.

3.2. The index of D. The virtual representation [V_{λ,C}] − [W_{λ,C}] ∈ R(G) is the same as Ind(D) = [Ker D] − [Coker D]. Furuta determines Ind(D) as a Pin (2) representation; denoting the restriction map r : R(G) → R(Pin (2)), Furuta shows

r(Ind(D)) = 2kh − m˜1

where k = −σ(X)/16 and m = b_{2}^{+}(X). Thus Ind(D) = sh − t˜1 where s and t are polynomials in ξ or ζ such that s(1) = 2k and t(1) = m. In the case of G = G_{ev} we write

s(ζ) = ∑_{i=1}^{2^{p}} s_{i}ζ^{i} and t(ζ) = ∑_{i=1}^{2^{p}} t_{i}ζ^{i}

and for G = G_{odd} we have

s(ξ) = ∑_{i=1}^{2^{p}} s_{i}ξ^{2i−1} and t(ξ) = ∑_{i=1}^{2^{p}} t_{i}ξ^{2i}.

To compute the coefficients t_{i} and s_{i} one can use the Z/2^{p}-index theorem. We will only need information about the t_{i} coefficients.

From the definition of the operator D we see that the polynomial ∑_{i=1}^{2^{p}} t_{i}ζ^{i} is the honest Z/2^{p} representation

Coker((d^{∗}, d^{+}) : Ω^{1} → Ω_{⊥}^{0} ⊕ Ω_{+}^{2})

where Ω_{⊥}^{0} denotes the L^{2}-orthogonal complement of the constant functions (this is true for both the odd and even cases). The constant coefficient, t_{2^{p}} is the dimension of the invariant part of H_{+}^{2}(X) which is just b_{2}^{+}(X/τ). More generally, b_{2}^{+} of the quotients of X by subgroups of Z/2^{p} are given by the various sums of the t_{i}’s. The dimension of the subspace of ∑_{i=1}^{2^{p}} t_{i}ξ^{i} invariant under Z/2^{j} ⊂ Z/2^{p} is given by

∑_{i≡0mod2^{j}} t_{i} = b_{2}^{+}(X_{j})

where X_{j} is the quotient of X by Z/2^{j} and X_{0} = X by convention.

We summarize the discussion in the following

Theorem 3.2. The index of D is given by

[V_{λ,C}] − [W_{λ,C}] = sh − t˜1 ∈ R(G)

where if G = G_{ev} then

s(ζ) = ∑_{i=1}^{2^{p}} s_{i}ζ^{i} and t(ζ) = ∑_{i=1}^{2^{p}} t_{i}ζ^{i}

and if G = G_{odd} then

s(ξ) = ∑_{i=1}^{2^{p}} s_{i}ξ^{2i−1} and t(ξ) = ∑_{i=1}^{2^{p}} t_{i}ξ^{2i}.

For either G_{ev} or G_{odd} we have

∑_{i=1}^{2^{p}} s_{i} = 2k
∑_{i≡0mod2^{j}} t_{i} = b_{2}^{+}(X_{j}).

3.3. The Thom isomorphism and a character formula for the K-theoretic degree. The Thom isomorphism theorem in equivariant K-theory for a general compact Lie group is a deep theorem proved using elliptic operators [1] . The subsequent character formula of this section uses only elementary properties of the Bott class. We follow tom Dieck [12] pgs. 254–255 for this discussion.

Let V and W be complex Γ representations for some compact Lie group Γ. Let BV and BW denote balls in V and W and let f : BV → BW be a Γ-map preserving the boundaries SV and SW. K_{Γ}(V) is by definition K_{Γ}(BV, SV) and by the equivariant Thom isomorphism theorem, K_{Γ}(V) is a free R(Γ) module with generator the Bott class λ(V). Applying the K-theory functor to f we get a map

f^{∗} : K_{Γ}(W) → K_{Γ}(V)

which defines a unique element α_{f} ∈ R(Γ) by the equation f^{∗}(λ(W)) = α_{f} · λ(V). The element α_{f} is called the K-theoretic degree of f.

Let V_{g} and W_{g} denote the subspaces of V and W fixed by an element g ∈ Γ and let V_{g}^{⊥} and W_{g}^{⊥} be the orthogonal complements. Let f^{g} : V_{g} → W_{g} be the restriction of f (well defined because of equivariance) and let d(f^{g}) denote the ordinary topological degree of f^{g} (by definition, d(f^{g}) = 0 if dim V_{g} ≠ dim W_{g}). For any β ∈ R(Γ) let λ_{−1}β denote the alternating sum ∑(−1)^{i}λ^{i}β of exterior powers.

T. tom Dieck proves a character formula for the degree α_{f}:

Theorem 3.3. Let f : BV → BW be a Γ-map preserving boundaries and let α_{f} ∈ R(Γ) be the K-theory degree. Then

tr_{g}(α_{f}) = d(f^{g}) tr_{g}(λ_{−1}(W_{g}^{⊥} − V_{g}^{⊥}))

where tr_{g} is the trace of the action of an element g ∈ Γ.

This formula will be especially useful in the case where dim W_{g} ≠ dim V_{g} so that d(f^{g}) = 0.

We also recall here that λ_{−1}(∑_{i} a_{i}r_{i}) = ∏_{i}(λ_{−1}r_{i})^{a_{i}} and that for a one dimensional representation r, we have λ_{−1}r = (1−r). A two dimensional representation such as h has λ_{−1}h = (1 − h + Λ^{2}h). In this case, since h comes from an SU(2) representation, Λ^{2}h = det h = 1 so λ_{−1}h = (2 − h).

All the proofs in the following subsections proceed by using the character formula to examine the K-theory degree α_{f_{λ}} of the map f_{λ} : BV_{λ,C} → BW_{λ,C} coming from the Seiberg-Witten equations. We will abbreviate α_{f_{λ}} as just α and V_{λ,C} and W_{λ,C} as just V and W and we will use the following elements of G. Let ϕ ∈ S^{1} ⊂ Pin (2) ⊂ G be an element generating a dense subgroup of S^{1}; let η ∈ Z/2^{p} and ν ∈ Z/2^{p+1} be generators and recall that there is the element J ∈ Pin (2) coming from the quaternions. Note that the action of J on h has two invariant subspaces on which J acts by multiplication with \sqrt{−1} and −\sqrt{−1}.

3.4. Proof of Furuta’s theorem (Theorem 1.1). Consider α = α_{f_{λ}} ∈ R(Pin (2)); it has the form

α = α_{0} + α̃_{0}˜1 + ∑_{i=1}^{∞} α_{i}h_{i}.

Since ϕ acts non-trivially on h and trivially on ˜1, we have that dim V_{ϕ} ≠ dim W_{ϕ} as long as m > 0. The character formula then gives

tr_{ϕ}(α) = 0 = α_{0} + α̃_{0} + ∑_{i=1}^{∞} α_{i}(ϕ^{i} + ϕ^{−i})

so α_{0} = −α̃_{0} and α_{i} = 0 for i ≥ 1.

Since J acts non-trivially on both h and ˜1, dim V_{J} = dim W_{J} = 0 so d(f^{J}) = 1 and the character formula says

tr_{J}(α) = tr_{J}(λ_{−1}(m˜1 − 2kh))
= tr_{J}((1 − ˜1)^{m}(2 − h)^{−2k})
= 2^{m−2k}

using tr_{J} h = 0 and tr_{J} ˜1 = −1. On the other hand tr_{J}(α) = tr_{J}(α_{0}(1 − ˜1)) = 2α_{0} so the degree is

α = 2^{m−2k−1}(1 − ˜1)

and we can conclude 2k + 1 ≤ m.

The proofs of Theorems 1.2, 1.3, and 1.4 are a generalization of the above proof.

3.5. Proof of Theorem 1.3. In this case the group is G = G_{ev} = Pin (2) × Z/2 and

V − W = (s_{1}ζ + s_{2})h − (t_{1}ζ + t_{2})˜1

where t_{1} + t_{2} = m and s_{1} + s_{2} = 2k. The hypothesis m ≠ b_{2}^{+}(X/σ) > 0 translates to t_{1} + t_{2} ≠ t_{2} > 0 which is equivalent to both t_{1} and t_{2} being non-zero.

Both ϕ and ϕν act non-trivially on h and trivially on ˜1 so (since t_{2} ≠ 0) we have d(f^{ϕ}) = d(f^{ϕν}) = 0 so that tr_{ϕ}(α) = tr_{ϕν}(α) = 0.

The general form of α is

α = α_{0}(ζ) + α̃_{0}(ζ)˜1 + ∑_{i=1}^{∞} α_{i}(ζ)h_{i}

so the conditions tr_{ϕ}(α) = tr_{ϕν}(α) = 0 imply that

α_{0}(±1) + α̃_{0}(±1) = 0
α_{i}(±1) = 0

since α_{0}, α̃_{0}, and α_{i} are degree 1 in ζ we see that α_{0} = −α̃_{0} and α_{i} = 0 so

α = (α_{0}^{1}ζ + α_{0}^{2})(1 − ˜1).

Now Jν acts trivially on ζ˜1 and non-trivially on ζh and h so since t_{1} ≠ 0, d(f^{Jν}) = 0. Thus we get

tr_{Jν}(α) = 0 = (−α_{0}^{1} + α_{0}^{2}) · 2

so that α_{0}^{1} = α_{0}^{2}.

Finally, as before, J acts non-trivially on W and V so that d(f^{J}) = 1 and tr_{J}(α) = 2^{m−2k} so it must be the case that

α = 2^{m−2k−2}(1 + ζ)(1 − ˜1)

and thus 2k + 2 ≤ m.

3.6. Proof of Theorem 1.4. We generalize the preceding proof. Now G = Pin (2)× (Z/2)^{q} and we write ζ_{i} for the non-trivial representation of the i-th copy of Z/2. The index Ind(D) is given by

V − W = s(ζ_{1}, …, ζ_{q})h − t(ζ_{1}, …, ζ_{q})˜1

where s and t are polynomial functions in the variables ζ_{1}, …, ζ_{q} of multi-degree (1, …, 1).

The hypothesis m ≠ b_{2}^{+}(X/g) then implies that the representation t(ζ_{1}, …, ζ_{q}) contains a summand on which g acts as −1. Since J acts on ˜1 by −1 we see that the representation t(ζ_{1}, …, ζ_{q})˜1 has a positive dimensional subspace fixed by Jg for every non-trivial g ∈ (Z/2)^{q}. On the other hand, Jg always acts non-trivially on s(ζ_{1}, …, ζ_{q})h so the character formula gives us

tr_{Jg}(α) = 0.

Since b_{2}^{+}(X/(Z/2)^{q}) ≠ 0, the coefficient of the trivial representation in t(ζ_{1}, …, ζ_{q}) is non-zero and so ϕg always fixes a non-trivial subspace of t˜1. On the other hand, ϕg always acts non-trivially on sh so again the character formula shows that

tr_{ϕg}(α) = 0.

The general form of α is

α = α_{0} + α̃_{0}˜1 + ∑_{i=1}^{∞} α_{i}h_{i}

where α_{0}, α̃_{0}, and α_{i} are polynomial functions in ζ_{1}, …, ζ_{q} of multi-degree (1, …, 1).

Now for any polynomial function β(x_{1}, …, x_{q}) of multi-degree (1, …, 1), if we know that β((−1)^{n_{1}}, …, (−1)^{n_{q}}) = 0 for any arrangement of the signs, then β ≡ 0. Thus the formula tr_{ϕg}(α) = 0 for all g ∈ (Z/2)^{q} implies that α_{0} + α̃_{0} ≡ 0 and α_{i} ≡ 0. We can therefore write

α = α_{0}(ζ_{1}, …, ζ_{q})(1 − ˜1).

Since J acts non-trivially on both U and V we can compute as before:

tr_{J}(α) = tr_{J}(λ_{−1}(W − V)) = 2^{m−2k}.

This equation, along with the 2^{q} − 1 equations tr_{Jg}(α) = 0 for g ≠ 1 ∈ (Z/2)^{q} give 2^{q} independent conditions on α_{0} which determine it uniquely. It must be the following:

α = 2^{m−2k−1−q}(1 + ζ_{1})(1 + ζ_{2}) ⋯ (1 + ζ_{q})(1 − ˜1)

and thus 2k + 1 + q ≤ m.

3.7. Proof of Theorem 1.2. Recall that

[V] − [W] = s(ξ)h − t(ξ)˜1 ∈ R(G_{odd})

with s(ξ) = ∑_{i=1}^{2^{p}} s_{i}ξ^{2i−1} and t(ξ) = ∑_{i=1}^{2^{p}} t_{i}ξ^{2i}. The K-theory degree α = α_{f_{λ}} of f_{λ} has the form

α = α_{0}(ξ) + α̃_{0}(ξ) + ∑_{i=1}^{∞} α_{i}(ξ)h_{i}.

We compute the following characters of α:

Lemma 3.4. tr_{ϕη^{j}}(α) = 0 for j = 1, …, 2^{p}, tr_{Jη^{j}}(α) = 0 for j = 1, …, 2^{p} − 1, and tr_{J}(α) = 2^{m−2k}.

Proof: We introduce the notation (V)_{g} = dim V_{g}. By the character formula, tr_{ϕη^{j}}(α) = 0 if

(V)_{ϕη^{j}} − (W)_{ϕη^{j}} = (s(ξ)h − t(ξ)˜1)_{ϕη^{j}} ≠ 0.

Since ϕη^{j} acts non-trivially on every ξ^{i}h and ϕη^{j} acts trivially on ˜1, we have

(V)_{ϕη^{j}} − (W)_{ϕη^{j}} ≤ −t_{2^{p}}.

By the hypothesis in Theorem 1.2, t_{2^{p}} = b_{2}^{+}(X/τ) > 0 so tr_{ϕη^{j}}(α) = 0.

To show tr_{Jη^{j}}(α) = 0 for j = 1, …, 2^{p} − 1 we need to show (V)_{Jη^{j}} − (W)_{Jη^{j}} ≠ 0. The 2 dimensional representation h decomposes into two complex lines on which J acts as \sqrt{−1} and −\sqrt{−1}, so

(ξ^{2i−1}h)_{Jη^{j}} = \begin{cases}1 & if η^{j} acts on ξ^{2i−1} by ±\sqrt{−1} \\ 0 & otherwise. \end{cases}

η^{j} acts as ±\sqrt{−1} on ξ^{2i−1} if and only if j and i satisfy

(2i − 1)j ≡ ±2^{p−1} mod 2^{p+1}.

If j = 2^{p−1} then the condition is satisfied for every i; if j ≠ 2^{p−1} the condition is never satisfied (divide both sides by the highest power of 2 that divides j to get an odd number on one side and an even on the other). Thus

(s(ξ)h)_{Jη^{j}} = \begin{cases}2k & if j = 2^{p−1}, \\ 0 & if j ≠ 2^{p−1}. \end{cases}

Now Jη^{j} acts on ˜1 by −1 and so

(ξ^{2i}˜1)_{Jη^{j}} = \begin{cases}1 & if η^{j} acts on ξ^{2i} by −1 and \\ 0 & otherwise. \end{cases}

Thus we see that

(t(ξ)˜1)_{Jη^{j}} = ∑_{2ij≡2^{p}mod2^{p+1}} t_{i}
= ∑_{ij≡2^{p−1}mod2^{p}} t_{i}.

We want to see that the non-degeneracy conditions of the theorem imply (V − W)_{Jη^{j}} ≠ 0. The conditions are equivalent to

0 < b_{2}^{+}(X_{p}) < b_{2}^{+}(X_{p−1}) < ⋯ < b_{2}^{+}(X_{1})

and

b_{2}^{+}(X_{1}) ≠ m − 2k

which in terms of the t_{i}’s are

t_{2^{p}} ≠ 0
t_{2^{p−1}} ≠ 0
t_{2^{p−2}} + t_{3·2^{p−2}} ≠ 0
t_{2^{p−3}} + t_{3·2^{p−3}} + t_{5·2^{p−3}} + t_{7·2^{p−3}} ≠ 0
⋮
t_{2} + t_{3·2} + t_{5·2} + ⋯ ≠ 0

and

∑_{i even} t_{i} ≠ m − 2k.

Since m = ∑ t_{i} the last condition is the same as

t_{1} + t_{3} + ⋯ + t_{2^{p}−1} ≠ 2k.

From the previous discussion we have in the case j ≠ 2^{p−1}

(V − W)_{Jη^{j}} = (s(ξ)h − t(ξ)˜1)_{Jη^{j}}
= − ∑_{ij≡2^{p−1}mod2^{p}} t_{i}
= −(t_{2^{a}} + t_{3·2^{a}} + ⋯)
≠ 0

where 2^{p−1−a} is the largest power of 2 dividing j.

For the case j = 2^{p−1}

(V − W)_{Jη^{j}} = 2k − ∑_{i2^{p−1}≡2^{p−1}mod2^{p}} t_{i}
= 2k − ∑_{i≡1mod2} t_{i}
≠ 0.

To complete the lemma we compute tr_{J}(α). Once again, since J acts non-trivially on W and V,

tr_{J}(α) = tr_{J} Λ_{−1}(W − V) = 2^{m−2k}.

Lemma 3.5. Let β(ξ) be of the form ∑_{i=1}^{2^{p}} β_{i}ξ^{2i} or ∑_{i=1}^{2^{p}} β_{i}ξ^{2i−1} in R(Z/2^{p+1}). If tr_{η^{j}} β = 0 for j = 1, …, 2^{p} then β ≡ 0.

Proof: Let \underline{β}(x) = ∑_{i=1}^{2^{p}} β_{i}x^{i}. \underline{β} is then a degree 2^{p} polynomial with roots at all of the 2^{p}-th roots of unity and at 0, so \underline{β} is identically 0.

The general form of α is

α = α_{0}(ξ) + α̃_{0}(ξ)˜1 + ∑_{i=1}^{∞} α_{i}(ξ)h_{i}

so using the computation of Lemma 3.4

0 = tr_{ϕη^{j}}(α)
= tr_{η^{j}}(α_{0} + α̃_{0}) + ∑_{i=1}^{∞} tr_{η^{j}}(α_{i})(ϕ^{i} + ϕ^{−i})

for j = 1, …, 2^{p} − 1.

Since each ϕ^{i} term must vanish separately, Lemma 3.5 immediately implies that α_{0} + α̃_{0} ≡ 0 and α_{i} ≡ 0. Thus α has the form α_{0}(ξ)(1 − ˜1) where α_{0}(ξ) = ∑_{i=1}^{2^{p}} α_{0}^{i}ξ^{2i}. Since tr_{Jη^{j}}(α) = tr_{η^{j}}(α_{0}) · 2 = 0 for j = 1, …, 2^{p} − 1, the degree 2^{p} polynomial

\underline{α}_{0}(x) = ∑_{i=1}^{2^{p}} α_{0}^{i}x^{i}

has 2^{p} known roots, namely the 2^{p} − 1 non-trivial 2^{p}-th roots of unity and 0. Thus \underline{α}_{0}(x) = const. ∑_{i=1}^{2^{p}} x^{i} and we can use tr_{J}(α) = 2^{m−2k} to determine the constant. Thus we can conclude that

α = 2^{m−2k−p−1}(1 + ξ^{2} + ξ^{4} + ⋯ + ξ^{2^{p+1}−2})(1 − ˜1),

and so 2k + 1 + p ≤ m and the theorem is proved.

Remark 3.1. For a spin action of an arbitrary group Γ of order 2^{p} on X our methods should give the bound 2k + 1 + p ≤ m as long as the action is subject to some non-degeneracy conditions. The proof would proceed as all the others, using the non-degeneracy conditions and the character formula to guarantee tr_{Jg}(α) = 0 for non-trivial g ∈ Γ and tr_{ϕg}(α) = 0 for all g ∈ Γ. Then one can show that

α = 2^{m−2k−1−p} · ρ · (1 − ˜1)

where ρ is the regular representation of Γ. Complications do occur: in order to guarantee that tr_{Jg}(α) = 0 one needs to incorporate information about the virtual representation s, i.e. the Γ-index of the Dirac operator. This can be computed via the index theorem, but in general the non-degeneracy conditions would then involve the fixed point set. A further complication occurs because the lift of the action to the spin bundle can be complicated; some group elements may lift with twice their original order while others may preserve their order. Nevertheless, our techniques could be applied in a case by case basis if the above issues are understood.

3.8. Proof of Theorem 1.5. This has a slightly different flavor than the previous proofs. The proof is essentially the same for G_{odd} and G_{ev}; we will use the G_{ev} notation. The index has the form

Ind(D) = s(ζ)h − t(ζ)˜1

and the hypothesis k > 0 means that s(ζ) = ∑_{i=1}^{2^{p}} s_{i}ζ^{i} has at least one positive coefficient. From Theorem 1.1, k > 0 also implies m > 0. We will prove that b_{2}^{+}(X/τ) ≠ 0 by contradiction. Suppose b_{2}^{+}(X/τ) = 0 so that the constant coefficient of t(ζ) is 0, i.e. t_{2^{p}} = 0. Since m > 0 we have (W − V)_{ϕ} ≠ 0 so tr_{ϕ}(α) = 0 and since t_{2^{p}} = 0, we know that ϕν acts non-trivially on all of V and W so

tr_{ϕν}(α) = λ_{−1}(W − V)
= \frac{(1 − ν)^{t_{1}}(1 − ν^{2})^{t_{2}} ⋯ (1 − ν^{2^{p}−1})^{t_{2^{p}−1}}}{(1 − νϕ)^{s_{1}}(1 − νϕ^{−1})^{s_{1}} ⋯} .

Since at least one of the s_{i}’s is positive, the above expression has arbitrarily high powers of ϕ in it. On the other hand

tr_{ϕν}(α) = tr_{ϕν} (α_{0}(ζ) + α̃_{0}(ζ)˜1 + ∑_{i=1}^{∞} α_{i}(ζ)h_{i})

= α_{0}(ν) + α̃_{0}(ν) + ∑_{i=1}^{∞} α_{i}(ν)(ϕ^{i} + ϕ^{−i})

has only finitely many non-zero α_{i} terms, which is our contradiction.

4. Applications

4.1. Genus bounds. Our original motivation for this work is our application to genus bounds. We refer the reader to our paper [4] for details of the set up.

Let M be a smooth 4-manifold (not necessarily spin) and let Σ ↪ M be a smoothly embedded surface representing a homology class a ∈ H_{2}(M; Z). A basic question in 4-manifold topology asks for the minimal genus g_{min}(a) of a smoothly embedded surface representing a given class a. In order to determine g_{min}(a) one needs good lower bounds on the genus of embedded surfaces and constructions realizing those bounds.

The basic ideas we use go back to Hsiang-Szczarba [13], Rochlin [10], and Kotschick-Matíc [7] (see also T. Lawson [8]).

Suppose the class a is divisible by some number d, then one can construct the dfold branched cover X → M branched along Σ. Under favorable hypotheses, X will be spin and the covering transformation is a spin action. The signature and Euler characteristic of X can be computed in terms of Σ · Σ and g(Σ) and so the bounds of Theorems 1.1 and 1.2 give genus bounds.

The most straightforward implementation of this idea is for double branched covers and results in Theorem 1.6. We will prove the general result coming from 2^{p} covers in this section.

Proposition 4.1. Let Σ ↪ M be a smoothly embedded surface of genus g in a smooth, simply-connected, compact, oriented 4-manifold. Suppose that [Σ] ∈ H_{2}(M) has the property that 2^{p}|[Σ] and [Σ]/2^{p} ≡ w_{2}(M) mod 2.

Then there is a spin 4-manifold X with a spin Z/2^{p} action τ : X → X of odd type such that X/τ = M. Furthermore, k = −σ(X)/16 and m = b_{2}^{+}(X) are given by:

k = \frac{1}{16} (−2^{p}σ(M) + \frac{4^{p} − 1}{3 · 2^{p}}[Σ] · [Σ]) ,
m = 2^{p}b_{2}^{+}(M) + (2^{p} − 1)g − \frac{4^{p} − 1}{6 · 2^{p}}[Σ] · [Σ].

Proof: This is in [4]; a sketch of the proof is the following. The condition 2^{p}|[Σ] allows one to construct the 2^{p}-fold branched cover X → M branched along Σ. One can compute w_{2}(X) using a formula of Brand [3] and the condition of [Σ]/2^{p} ≡ w_{2}(M) mod 2 guarantees w_{2}(X) vanishes. π_{1}(X) is finite (see [10], [7]) so b_{1}(X) = b_{3}(X) = 0 and so m and k can be determined by e(X) and σ(X) which can be computed by the G-index theorem. The covering action τ : X → X is automatically spin because H_{1}(X; Z/2) = 0 (see [7]) and the action is of odd type since the fixed point set is two dimensional.

Here is our general genus bound (Theorem 1.6 is a special case):

Theorem 4.2. Let Σ ↪ M be a smoothly embedded surface of genus g in a smooth, simply-connected, compact, oriented 4-manifold. Suppose that [Σ] ∈ H_{2}(M) has the property that 2^{p}|[Σ] and [Σ]/2^{p} ≡ w_{2}(M) mod 2.

Suppose b_{2}^{+}(M) > 1 and g ≠ [Σ]^{2}(1 + 2^{2i−2p+1})/6 − b_{2}^{+}(M) for i = 1, …, p − 1. Then

g ≥ \frac{1}{2^{p} − 1} [\frac{5}{4} (\frac{4^{p} − 1}{6 · 2^{p}}[Σ]^{2} − 2^{p−1}σ(M)) + 1 + p − 2^{p−1}b_{2}(M)] . (2)

Remark 4.1. The conditions on b_{2}^{+}(M) and g are so that the non-degeneracy conditions of Theorem 1.2 are met. The condition on g is not present for p = 1. Although it is aesthetically unpleasing to have restrictions on g in the hypothesis, in practice the conditions are easy to dispense with (see example 4.1 ). Even without the conditions one still gets an inequality coming from Furuta’s theorem which is the same as the bound (2) but without the p term in the brackets. Our theorem thus improves the Furuta bound by p/(2^{p} − 1). Probably the best applications occur in the case p = 1 (as in Theorem 1.6 where we improve the Furuta bound by 1). However, we will give examples where Theorem 4.2 provides sharp bounds unobtainable from the Furuta bound.

Example 4.1. Let M = #_{N}CP^{2} with N > 1 and consider Σ ↪ M representing the class (4, …, 4). We can apply Theorem 4.2 with p = 2. It tells us

g ≥ \frac{1}{3}(8N + 3)

if g ≠ 3N. When N ≥ 6 the condition on g is no condition and for N = 2, …, 5 the theorem gives g ≥ 3N since if g = 3N − 1 it would contradict g ≥ \frac{1}{3}(8N + 3). Since the connected sum of the algebraic representative in each factor has genus 3N, the theorem gives a sharp bound for N = 2, …, 5.

Proof of Theorem 4.2 : Let X → M be the 2^{p}-th branched cover and, to match the notation of Theorem 1.2, let X_{i} be the 2^{p−i} branched cover. Then X_{0} = X, X_{p} = M and X_{i} is the quotient of X by Z/2^{i} ⊂ Z/2^{p}. Let m_{i} = b_{2}^{+}(X_{i}) which is computable by essentially the same formula as in Proposition 4.1

m_{i} = 2^{p−i}m_{p} + (2^{p−i} − 1)g − \frac{4^{p−i} − 1}{6 · 2^{p−i}}[Σ]^{2}.

Note that m_{0} ≥ m_{1} ≥ ⋯ ≥ m_{p} > 1. The inequality of the theorem is equivalent to 2k + 1 + p ≤ m_{0} so the only way the theorem can fail is if 2k + δ = m_{0} for δ ∈ {1, …, p} and m_{0} − 2k = m_{1} or m_{i} = m_{i−1} for i = 2, …, p. The hypothesis g ≠ [Σ]^{2}(1 + 2^{2i−2p+1})/6 − b_{2}^{+}(M) is equivalent to m_{i} ≠ m_{i+1} so we know that m_{1} > m_{2} > ⋯ > m_{p} = b_{2}^{+}(M). This implies m_{1} ≥ p − 1 + b_{2}^{+}(M) which, combined with the hypothesis b_{2}^{+}(M) > 1 gives m_{1} > p. Then we see that we cannot have 2k + δ = m_{0} and m_{0} − 2k = m_{1} since that would imply δ > p.

4.2. Involutions on rational cohomology K3’s. The proof of Theorem 1.8 follows readily from our main theorems (Theorems 1.2, 1.3, and 1.5). Suppose X is a rational cohomology K3 so that H^{∗}(X; Q) ≅ H^{∗}(K3; Q) and suppose τ : X → X is a spin involution.

First suppose that τ is of odd type. Since the inequality of Theorem 1.2 is violated, one of the non-degeneracy conditions must also be violated and so b_{2}^{+}(X/τ) is either 1 or 0. Theorem 1.5 shows that b_{2}^{+}(X/τ) is not 0 so it must be 1. This can easily occur, for example K3 is a double branched cover over CP^{2}, S^{2} × S^{2}, and E(1) ≅ CP^{2}#9\overline{CP}^{2} and it has a free involution covering the Enriques surface.

Now suppose that τ is even so it must violate the non-degeneracy conditions of Theorem 1.3. In conjunction with Theorem 1.5 we see that b_{2}^{+}(X/τ) must be 3. Since the fixed point set is at most points, the G-signature theorem tells us that

σ(X/τ) = \frac{1}{2}σ(X) = −8.

If N is then the number of fixed points, then the Lefschetz formula tells us the Euler characteristic

χ(X/τ) = \frac{1}{2}(χ(X) + N) = 12 + N/2.

Since b_{1}(X/τ) = 0 and we know b_{2}^{+}(X/τ) = 3, we can solve the above equations to get b_{2}^{−}(X/τ) = 11 and N = 8.

It was pointed out to us by P. Kronheimer that this does occur. The construction is as follows. Let Ỹ be the K3 surface. There are 8 disjoint −2 spheres S_{1}, …, S_{8} such that S = ∑_{i=1}^{8} S_{i} is divisible by 2. Let X̃ be the double branched cover of Ỹ branched along S. The preimage of S in X̃ is then 8 disjoint (−1) spheres which we can blow down to obtain X, a smooth manifold with an involution fixing 8 points covering Y, the orbifold obtained by collapsing the S_{i}’s in Ỹ:

\begin{matrix}X̃ & ⟶ & X \\ ↓ & & ↓ \\ Ỹ & ⟶ & Y. \end{matrix}

We will show that X is a rational cohomology K3 (with a little more work, one can see that X ≅ K3) and b_{2}^{+}(Y) = 3 so this is the example we seek. From the signature theorem and Lefschetz formula we compute that χ(X̃) = 2χ(Ỹ) − χ(S) = 32 and σ(X̃) = 2σ(Ỹ) − S^{2} = 24 and so, after blowing down, χ(X) = 24 and σ(X) = −16. Since the preimage of S is characteristic in X̃ we know that X is spin and since b_{1}(X̃) = b_{1}(X) = 0, X is a rational cohomology K3.

Remark 4.2. It is easy to construct a homeomorphism τ : X → X generating an involution with b_{2}^{+}(X/τ) = 2 (for example) but our classification implies this will not be smoothable, i.e. there is no smooth structure on X so that this τ is a diffeomorphism.

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[3] N. Brand. Necesary conditions for the existence of branched coverings. Inventiones Math., 54, 1979.

[4] Jim Bryan. Seiberg-Witten à la Furuta and genus bounds for classes with divisibility. In S. Akbulut and T. Onder, editors, Proceedings of the 5^{th} Gökava Geometry and Topology conference, 1996.

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[8] T. Lawson. The minimal genus problem. Preprint.

[9] J. W. Morgan, Z. Szabó, and C. H. Taubes. The generalized Thom conjecture. Preprint.

[10] Rokhlin. Two-dimensional submanifolds of four-dimensional manifolds. Functional Analysis and Applications., 6:93–48, 1971.

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[12] Tammo tom Dieck. Transformation Groups and Representation Theory. Number 766 in Lecture Notes in Mathematics. Springer-Verlag, 1970.

[13] W.C.Hsiang and R.H.Szczarba. On embedding surfaces in four-manifolds. Proceedings of the Symposium of Pure Math, 22:97–103, 1971.

M.S.R.I., 1000 Centennial Dr., Berkeley, CA 94720

E-mail address: jbryan@msri.org