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Smoothability of Z\times Z-actions on 4-manifolds

We construct a nonsmoothable Z\times Z-action on the connected sum of an Enriques surface and S^2\times S^2, such that each of generators is smoothable. We also construct a nonsmoothable self-homeomorphism on an Enriques surface.


Introduction
The purpose of this paper is to prove the existence of a nonsmoothable Z × Z-action on a 4-manifold, such that each of the generators is smoothable: Theorem 1.1. Let X be the connected sum of an Enriques surface with S 2 × S 2 . Then, there exists a pair (f 1 , f 2 ) of self-homeomorphisms of X which has the following properties: (1) f 1 and f 2 commute.
(2) Each one of f 1 and f 2 can be smoothed for some smooth structure on X. However, f 1 and f 2 can not be smoothed at the same time for any smooth structure on X.
We also construct a nonsmoothable self-homeomorphism of an Enriques surface.
Theorem 1.2. There exists a self-homeomorphism of an Enriques surface Y which is nonsmoothable with respect to any smooth structure on Y .
To prove these results, we modify the argument in [5] which analyses the Seiberg-Witten moduli for families, and give more convenient constraints on diffeomorphisms of 4-manifolds, and then, construct homeomorphisms which violate the constraints.
Acknowledgements. The author would like to thank M. Furuta, Y. Kametani, K. Kiyono and M. Ue for helpful discussions and comments on earlier versions of the paper. It is also his pleasure to thank the referee for detailed comments and many valuable suggestions which enable him to make the paper corrected and improved.

Constraints on diffeomorphisms
In this section, we review the paper [5], and give some modifications of its results. In the paper [5], the author investigated the Seiberg-Witten moduli of families of 4-manifolds, and as an application, gave some constraints on diffeomorphisms of 4-manifolds. Let X be a closed oriented smooth 4-manifold, and B another closed manifold. We assume a family X of X over B is given as a fiber bundle over B whose fibers are diffeomorphic to X as oriented manifolds. The fiber over b ∈ B is denoted by X b . Let T (X/B) be the tangent bundle along the fiber of X, and assume a metric on T (X/B) is given. In order to consider the Seiberg-Witten equations on the family X, we need a family of Spin c -structures on X.
One can obtain such a family of Spin c -structures if a Spin c -structure on T (X/B) is given. For this purpose, we gave somewhat complicated sufficient conditions. (See Proposition 2.1 of [5] and its correction [6].) In order to obtain a more convenient condition, we will take an alternative approach using classifying maps as described in [7].
Let Diff(X) be the group of orientation-preserving diffeomorphisms of X. The classifying space B Diff(X) classifies families X → B as above. Suppose a Spin c -structure c on X is given. Let us consider the group S(X, c) of pairs (f, u), where f is an orientation-preserving diffeomorphism and u : f * c → c is an isomorphism. The corresponding classifying space BS(X, c) classifies families X → B with a Spin c -structurec on T (X/B) such that the restriction ofc to each fiber is isomorphic to c. We have the forgetful map Φ : S(X, c) → Diff(X). In general, Φ is not surjective. Let N (X, c) be the image of Φ. Then there is an exact sequence where G = Aut(c) ∼ = Map(X, S 1 ). Note that BN (X, c) classifies families X → B whose structure groups are included in N (X, c). The exact sequence leads to a fibration Suppose it is given a family X → B classified by ρ : Two kinds of families whose structure groups are in N (X, c) will be used in the proofs of propositions below. The first is a mapping torus X f = (X × [0, 1])/f → S 1 defined by a diffeomorphism f : X → X satisfying f * c ∼ = c. The second is a "double" mapping torus X (f 1 ,f 2 ) → S 1 × S 1 defined by two commutative diffeomorphisms f 1 and f 2 satisfying If the family X is X f or X (f 1 ,f 2 ) as above, we always have a Spin c -structure on T (X/B) by the previous paragraph.
When a Spin c -structurec on T (X/B) is given, the Seiberg-Witten moduli space for the family X is given as follows. Let us define the bundle of parameters Π → B by where Met(X b ) is the space of Riemannian metrics on X b and * b is the Hodge star for the metric g b . If we choose a section η of Π, then the moduli space for the family (X,c) is defined by With these understood, we can modify the results in [5] as follows. For a Spin c -structure c on X, let L be the determinant line bundle of c. Then the virtual dimension d(c) of the Seiberg-Witten moduli of (X, c) is given by, Proposition 2.1. Let X be a closed oriented smooth 4-manifold with b 1 = 0 and b + = 1, c a Spin c -structure on X with d(c) = 0, and f : X → X an orientation-preserving diffeomorphism. If f * c is isomorphic to c, then f preserves the orientation of H + (X; R).
The proof of Proposition 2.1 is given by a slight modification of the proof of Theorem 1.2 of [5]. For reader's convenience, we outline it briefly. Suppose a diffeomorphism f satisfying f * c ∼ = c is given, and consider the mapping torus X f → B = S 1 by f . Under the assumptions of Proposition 2.1, the moduli space M(X f ,c, η) of X f for a generic choice of η is a compact 1-dimensional manifold whose boundary points consist of reducibles. Let us introduce a vector bundle (Such a bundle H η is defined for any family.) Then, it is proved the number of reducibles is equal modulo 2 to the number of zeros of a generic section of H + η . If f reverses the orientation of H + (X; R), then H + η is a nontrivial real line bundle over S 1 . This is a contradiction, because the number of boundary points of a compact 1-dimensional manifold is even.
Similarly, we can prove the following by modifying the proof of Theorem 1.1 of [5]: Suppose a pair (f 1 , f 2 ) of orientation preserving diffeomorphisms on X satisfies the following conditions: (1) f 1 and f 2 commute.
(2) f 1 and f 2 preserve the isomorphism class of c.

Nonsmoothable self-homeomorphism on Enriques surface
The purpose of this section is to prove Theorem 1.2. First, note that the Enriques surface can be decomposed into three connected summands topologically by a theorem due to Hambleton and Kreck [3]. In fact, the following theorem can be proved from Theorem 3 in [3] and its proof. [3]). The Enriques surface is homeomorphic to a topological manifold Y = |E 8 |#Σ#(S 2 × S 2 ), where |E 8 | is the "E 8 -manifold", i.e., the simplyconnected closed topological 4-manifold whose intersection form is the negative definite E 8 , and Σ is a non-spin rational homology 4-sphere with fundamental group Z/2. Now, we will construct a self-homeomorphism of Y . Let ϕ : S 2 × S 2 → S 2 × S 2 be an orientation-preserving diffeomorphism which has the following properties: (1) There is a 4-ball B 0 ⊂ S 2 × S 2 such that the restriction of ϕ to B 0 is the identity map on B 0 . (2) ϕ reverses the orientation of H + (S 2 × S 2 ; R). Such a ϕ can be easily constructed as follows: Let ϕ 0 be the automorphism on CP 1 × CP 1 defined by the complex conjugation. Choose a fixed point p 0 of ϕ 0 . Then, a required ϕ is obtained by perturbing ϕ 0 around p 0 to be the identity on a neighborhood of p 0 .
Let us define a self-homeomorphism f on Y by f = id |E 8 |#Σ #ϕ, where id |E 8 |#Σ is the identity map of |E 8 |#Σ. (Note that we can take a connected sum of ϕ with id |E 8 |#Σ on B 0 ⊂ S 2 × S 2 .) Now, we claim that f is nonsmoothable with respect to any smooth structure on Y .
To prove f nonsmoothable, we will temporarily need a topological Spin c -structure on the topological manifold Y . Let us make a digression for it. (A brief description for topological spin structures is found in [1], Section 3. See also [2], 10.2B.) By Kister-Mazur's theorem, the tangent microbundle τ Y determines up to isomorphism the topological "frame" bundle F whose structure group STop(4) consists of orientation-preserving homeomorphisms of R 4 preserving the origin. It is known that the inclusion SO(4) → STop(4) induces an isomorphism on π 1 and both have trivial π 0 and π 2 ([4],V and [2], 8.7). Let φ : SpinTop(4) → STop(4) be the unique double covering. Then, a topological spin structure on Y is defined as a double coveringF → F whose restriction to each fiber is φ. Topological Spin cstructures are similarly defined by using SpinTop c (4) := SpinTop(4) × Z 2 U(1) → STop(4). The set of isomorphic classes of topological Spin c -structures has a principal action of H 2 (Y ; Z) as in the case of true Spin c -structures. Proof. In this proof, all spin/Spin c -structures are understood as topological ones. The Spin c -structure c can be identified with the sum of the unique spin structure c 0 on |E 8 |#(S 2 × S 2 ) and a Spin c -structure c Σ on Σ whose c 1 (L) is a torsion class. Since f is the identity on Σ, f preserves c Σ . On the other hand, since c 0 is the unique spin structure on |E 8 |#( Let us prove f nonsmoothable. Once a smooth structure on Y is given, we have a reduction of the topological frame bundle F to the true frame SO(4)-bundle, and also a topological Spin c -structure is reduced to the corresponding true Spin c -structure. Suppose f is smoothed. By Lemma 3.4, f * c is isomorphic to c as true Spin c -structures. On the other hand, f is an orientation-preserving diffeomorphism which reverses the orientation of H + (Y ). This contradicts Proposition 2.1.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1. To begin with, we collect the ingredients needed for our construction. Let S 0 = S 2 × S 2 , and fix a 4-ball B ′ 0 ⊂ S 0 . For i = 1, 2, let (S i , ϕ i ) be copies of (S 2 × S 2 , ϕ), and fix smooth 4-balls B ′ i ⊂ S i on which ϕ i | B ′ i are the identity maps. If we make a connected sum of S i (i = 0, 1, 2) with another manifold, remove B ′ i from S i and glue it along the boundary to another. Let Z be |E 8 |#Σ. Later, we will choose 4-balls B 0 , B 1 and B 2 in Z so that • if we make a connected sum of Z with S i (i = 0, 1, 2), remove B i from Z and glue Z \ B i and S i \ B ′ i . (The resulting connected sum will be denoted as Z# ∂B i S i .) Let E 1 and E 2 be smooth 4-manifolds homeomorphic to an Enriques surface. The basic idea of our construction is as follows. The connected sum S 1 # ∂B 1 Z# ∂B 2 S 2 can be assumed as a connected sum of an Enriques surface with S 2 × S 2 in two ways: S 1 #E 1 and E 2 #S 2 . Then, commutative two homeomorphisms f 1 , f 2 will be defined by ϕ 1 # id E 1 and id E 2 #ϕ 2 , Let us begin the precise construction. Choose a 4-ball B 0 ⊂ Z arbitrarily. Then Z# ∂B 0 S 0 is homeomorphic to an Enriques surface. Fix a homeomorphismh 1 : Note that S 1 # ∂B 1 Z is also homeomorphic to an Enriques surface. Fix a homeomor- is a smoothly embedded 4-ball in E 2 . Take a smooth connected sum E 2 # ∂D 2 S 2 and a (topological) connected sum 1 is the identity map. Via homeomorphisms h, h 1 and h 2 , we obtain selfhomeomorphisms f 1 and f 2 of X := S 1 # ∂B 1 Z# ∂B 2 S 2 induced fromf 1 andf 2 , respectively. Then each f i (i = 1, 2) is smoothable for the smooth structure E i # ∂D i S i . Clearly, f 1 and f 2 commute. Let c be the Spin c -structure on X whose c 1 (L) is a torsion class. As in Lemma 3.4, we can see that f 1 and f 2 preserve the isomorphism class of c. However, w 2 H + η = 0 by construction. By Proposition 2.2, f 1 and f 2 can not be smoothed at the same time. Thus, Theorem 1.1 is proved.

Remarks
We give two remarks. The first is on another possibility of application of Proposition 2.2. The following problem would be interesting: Find two diffeomorphisms of a smooth manifold homeomorphic to a connected sum of an Enriques surface E with S 2 × S 2 that are simultaneously smoothable, commute up to isotopy, but do not have representatives in their isotopy classes that commute. If we want to construct such two diffeomorphisms on the smooth connected sum E#S 2 × S 2 , then one of the difficulties would be as follows. To appeal to Proposition 2.2, one of two diffeomorphisms will be required to reverse the orientation of the H + (E)-part of H 2 (E#S 2 × S 2 ), and it will be easy if we can construct such a diffeomorphism as a connected sum of a diffeomorphism f of E with one of S 2 × S 2 . However, this method is impossible, because Proposition 2.1 prohibits such an f . The second remark is on a generalization of the construction of the moduli spaces for families. In fact, we can construct the moduli space for a family without a family of Spin cstructures. More precisely, we claim the following: When a family X → B is classified by ρ : B → BN (X, c), we can always construct the moduli space M(X, c) for the family X, even if ρ does not have a liftρ : B → BS(X, c). The construction is outlined as follows. By taking local trivializations, the family X can be given via transition functions ψ βα : U α ∩ U β → N (X, c) for an appropriate covering {U λ } λ∈Λ of X. Suppose the intersection of every two members in {U λ } λ∈Λ is contractible. Then we can take a lift of each ψ βα : U α ∩ U β → N (X, c) toψ βα : U α ∩ U β → S(X, c). In general, suchψ βα do not satisfy the cocycle condition, but satisfy it up to gauge, i.e., ψ γβ ψ βα ψ −1 γα is a gauge transformation. One can define local families M(U λ × X, c) = b∈U λ M({b} × X, c) → U λ of moduli spaces and attaching mapsψ * βα between them induced fromψ βα . (Here, we need a little care on metrics and perturbations.) Since the moduli spaces are defined as the quotient spaces divided by the gauge transformations,ψ * βα satisfy the cocycle condition. Therefore, the global family M(X, c) can be constructed from the local families M(U λ × X, c) viaψ * βα . Such a family M(X, c) would be useful for further applications.