2.1 Mod 2 basic classes in $H^2(M;\mathbb {Z})/\operatorname {\mathrm {Aut}}(H^2(M;\mathbb {Z}))$

A building block of the construction of homology classes of $B\mathrm {Diff}^+(X)$ is a 4-manifold M that admits infinitely many exotic structures. This is inspired by Ruberman’s argument [Reference Ruberman31] in his work on Torelli groups. (See also Auckly’s recent work [Reference Auckly2] for one version of Ruberman’s argument in a Seiberg–Witten context.) Compared with [Reference Ruberman31, Reference Auckly2], what we newly need to require is that those exotic structures are distinguished by mod 2 basic classes that are distinct in $H^2(M;\mathbb {Z})/\operatorname {\mathrm {Aut}}(H^2(M;\mathbb {Z}))$ , the quotient of $H^2(M;\mathbb {Z})$ by the automorphism group of the intersection form. This is a reflection that we shall consider the whole diffeomorphism group, rather than the Torelli diffeomorphism group.

To formulate this precisely, let us introduce some notation. Let M be a smooth simply-connected closed oriented 4-manifold with $b^+(M) \geq 2$ . Since $H^2(M;\mathbb {Z})$ has no torsion, we can identify a spin $^c$ structure on M with a characteristic element in $H^2(M;\mathbb {Z})$ . Recall that a characteristic element ${c \in H^2(M;\mathbb {Z})}$ is called a basic class if $SW(M,c)$ , the Seiberg–Witten invariant, is nonzero. If $SW(M,c) \neq 0$ mod $2$ , we say that c is a mod 2 basic class. For simplicity, whenever we say that c is a (mod 2) basic class, we further impose that the formal dimension of c is zero (see (5)). We denote by $\mathcal {B}_2(M)$ the set of mod 2 basic classes of M. Note that $\mathcal {B}_2(M)$ is preserved under the $\mathbb {Z}/2$ -action on $H^2(M;\mathbb {Z})$ via multiplication by $-1$ . For a nonzero cohomology class $x \in H^2(M;\mathbb {Z})$ , let $\mathrm {div}(x)$ denote the divisibility of x – namely,

$$\begin{align*}\mathrm{div}(x) = \max\left\{n \in \mathbb{Z} \mid \exists y \in H^2(M;\mathbb{Z}) \text{ such that } ny=x\right\}\!. \end{align*}$$

For the zero element, we formally set $\mathrm {div}(0)=0$ in this paper. For a characteristic element $c \in H^2(M;\mathbb {Z})$ , define

$$\begin{align*}N(M;c) = \#\{[x] \in \mathcal{B}_2(M)/(\mathbb{Z}/2) \mid \mathrm{div}(x)=\mathrm{div}(c),\ x^2=c^2\}, \end{align*}$$

where $x^2$ denotes the self-intersection of x. In this section, we consider a 4-manifold M to satisfy the following assumption:

It is worth adding notes on the last property (iii) of Assumption 2.1. The principal intention of (iii) is to ensure that the mod 2 basic classes are distinct even in the quotient $H^2(M;\mathbb {Z})/\operatorname {\mathrm {Aut}}(H^2(M;\mathbb {Z}))$ (after passing to a subsequence, if necessary). For most 4-manifolds, increasing either divisibilities or self-intersections is the only possible way to get infinitely many characteristics distinct in $H^2(M;\mathbb {Z})/\operatorname {\mathrm {Aut}}(H^2(M;\mathbb {Z}))$ (cf. Proposition 4.5). The reason why we suppose $N(M_i; c_i)$ is odd is that we want to control sums of mod 2 Seiberg–Witten invariants over some class of spin $^c$ structures.

As a series of examples of M satisfying Assumption 2.1, we have the following:

To see Lemma 2.2, let us consider logarithmic transformations. For $n \geq 2$ and $i \geq 1$ , let $E(n;i)$ denote the logarithmic transformation of order $i>0$ performed on $E(n)$ (i.e., $E(n;i)$ is the elliptic surface of degree n with a single multiple fiber of order i). (Note that $E(n;1)=E(n)$ .) The Seiberg–Witten invariants of $E(n;i)$ were computed by Fintushel–Stern [Reference Fintushel and Stern12]. For readers’ convenience, we recall the result following their survey [Reference Fintushel and Stern13, Lecture 2].

In general, let Z be an oriented closed smooth 4-manifold with $b^+(Z) \geq 2$ , without torsion in $H^2(Z;\mathbb {Z})$ . Consider the Laurent polynomial

$$\begin{align*}\mathcal{SW}_Z = \sum_{c}SW(Z,c)t_c. \end{align*}$$

Here, $c \in H^2(Z;\mathbb {Z})$ are characteristic elements and $t_c$ are formal variables in $\mathbb {Z}[H^2(Z;\mathbb {Z})]$ corresponding to c. Note that $t_{c}t_{c'}=t_{c+c'}$ – in particular, $t_c^m=t_{mc}$ for $m \in \mathbb {Z}$ .

Now consider $Z=E(n;i)$ . Let $F \in H_2(E(n;i);\mathbb {Z})$ be the class that represents a generic fiber of the elliptic fibration. The multiple fiber of $E(n;i)$ represents a primitive homology class, which is given by $F/i$ . Let $F_i$ denote the Poincaré dual of $F/i$ and set $t = t_{F_i}$ . Then the Seiberg–Witten polynomial for $E(n;i)$ is given by

(2) $$ \begin{align} \mathcal{SW}_{E(n;i)} = (t^{i}-t^{-i})^{n-2}(t^{i-1} + t^{i-3} + \dots + t^{1-i}). \end{align} $$

2.2 Families over the torus

Fix $k>0$ , and let us take a 4-manifold M satisfying Assumption 2.1. Fix a diffeomorphism $M_i\# S^2 \times S^2 \to M\# S^2 \times S^2$ and identify $M_i\# kS^2 \times S^2$ with X for every i. Set $X = M\#kS^2 \times S^2$ .

We recall a construction of a smooth fiber bundle over $T^k$ with fiber X considered in [Reference Konno19, Reference Konno and Lin20]. Define an orientation-preserving diffeomorphism $f_0 : S^2 \times S^2 \to S^2 \times S^2$ by $f_0(x,y) = (r(x), r(y))$ , where $r : S^2 \to S^2$ is the reflection about the equator. By isotoping $f_0$ , we can obtain a diffeomorphism $f : S^2 \times S^2 \to S^2 \times S^2$ that fixes a disk $D^4 \subset S^2 \times S^2$ pointwise. Take copies $f_1, \ldots , f_k$ of f, and implant them into $M_i\# kS^2\times S^2$ for each i, by extending by the identity. Thus, we obtain diffeomorphisms $f_1, \ldots , f_k : M_i\# kS^2\times S^2 \to M_i\# kS^2\times S^2$ . Since the supports of $f_1, \ldots , f_k$ are mutually disjoint, and $f_1, \ldots , f_k$ commute each other. Using these commuting diffeomorphisms, we can form the multiple mapping torus $E_i \to T^k$ , which is a smooth fiber bundle with fiber $M_i\# kS^2\times S^2$ . Using the fixed identification between $M_i\# kS^2\times S^2$ and X, we obtain smooth fiber bundles (denoted by the same notation) $X \to E_i \to T^k$ with fiber X. Since f is orientation-preserving, the resulting fiber bundles $E_i$ are oriented (i.e., the structure group reduces to $\mathrm {Diff}^+(X)$ , the orientation-preserving diffeomorphism group).

For each $i\geq 1$ , regard $E_i$ as a continuous map $E_i : T^k \to B\mathrm {Diff}^+(X)$ . Now we set

(3) $$ \begin{align} \alpha_i := (E_1)_\ast([T^k]) - (E_i)_\ast([T^k]) \in H_k(B\mathrm{Diff}^+(X);\mathbb{Z}). \end{align} $$

This construction of the homology class $\alpha _i$ is the same as the one in [Reference Konno and Lin20, Proof of Theorem 3.10], except only for a condition on the Seiberg–Witten invariants of 4-manifolds. The origin of this construction is the first examples of exotic diffeomorphisms of 4-manifolds due to Ruberman [Reference Ruberman30].

Let us observe a few properties of $\alpha _i$ :

Lemma 2.4. The homology class $\alpha _i$ lies in

$$\begin{align*}\ker(i_\ast : H_k(B\mathrm{Diff}^+(X);\mathbb{Z}) \to H_k(B\mathrm{Homeo}^+(X);\mathbb{Z})). \end{align*}$$

For each i, fix a smoothly embedded 4-disk $D^4_i \subset M_i$ and similarly take $D^4 \subset M$ . Set $\mathring {M}_i = M_i \setminus \mathrm {Int}(D^4_i)$ and $\mathring {M} = M \setminus \mathrm {Int}(D^4)$ . We can find a diffeomorphism $\psi _i : M_i\#S^2\times S^2 \to M\#S^2\times S^2$ with $\psi _i(D^4_i)=D^4$ that respect parametrizations of $D^4_i$ and $D^4$ . Thus, we can identify $\mathring {M}_i\#S^2 \times S^2$ with $\mathring {M}\#S^2 \times S^2$ . Using these identifications, the construction of $E_i$ can be carried out with fixing the 4-disks, so we also have a homology class $\mathring {\alpha }_i \in H_k(B\mathrm {Diff}_\partial (\mathring {X});\mathbb {Z})$ defined similarly to $\alpha _i$ – namely,

$$ \begin{align*} \mathring{\alpha}_i := (E_1)_\ast([T^k]) - (E_i)_\ast([T^k]) \in H_k(B\mathrm{Diff}_\partial(\mathring{X});\mathbb{Z}), \end{align*} $$

where $E_i$ are regarded as maps $E_i : T^k \to B\text {Diff}_\partial (\mathring {X})$ . Letting $\rho : \mathrm {Diff}_\partial (\mathring {X}) \to \mathrm {Diff}^+(X)$ be the extension map by the identity, we have that $\alpha _i$ is the image of $\mathring {\alpha }_i$ under the induced map

$$\begin{align*}\rho_\ast : H_k(B\mathrm{Diff}_\partial(\mathring{X});\mathbb{Z}) \to H_k(B\mathrm{Diff}^+(X)\mathbb{Z}). \end{align*}$$

Lemma 2.5. The homology class $\mathring {\alpha }_i$ lies in the kernel of

$$\begin{align*}s_\ast : H_k(B\mathrm{Diff}_\partial(\mathring{X});\mathbb{Z}) \to H_k(B\mathrm{Diff}_\partial(\mathring{X}\# S^2 \times S^2);\mathbb{Z}). \end{align*}$$

The following is the most general statement of this paper:

To prove Theorem 2.7, due to Lemmas 2.4 to 2.6, it suffices to show that there is a homomorphism

$$\begin{align*}H_k(B\mathrm{Diff}_\partial(\mathring{X});\mathbb{Z}) \to (\mathbb{Z}/2)^{\infty} \end{align*}$$

that restricts to a surjection

$$\begin{align*}\left<\mathring{\alpha}_i \mid i\geq2\right> \twoheadrightarrow (\mathbb{Z}/2)^{\infty}. \end{align*}$$

We shall prove this remaining part in the subsequent sections.

Assuming Theorem 2.7, we obtain the proofs of results exhibited in Section 1:

3.1 Output

The proof of Theorem 2.7 uses characteristic classes defined by using Seiberg–Witten theory. Fix $k\geq 0$ , and let X be a smooth closed oriented 4-manifold with $b^+(X) \geq k+2$ . Lin and the author [Reference Konno and Lin20] defined a characteristic class

(4) $$ \begin{align} \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X) \in H^k(B\mathrm{Diff}^+(X);\mathbb{Z}/2), \end{align} $$

which we called the half-total Seiberg–Witten characteristic class. This was inspired by Ruberman’s total Seiberg–Witten invariant of diffeomorphisms [Reference Ruberman32], together with a gauge-theoretic construction of characteristic classes by the author [Reference Konno19]. We introduce generalizations of the characteristic class (4) to prove Theorem 2.7 as follows.

Let $\mathrm {Spin}^{c}(X,k)$ denote the set of isomorphism classes of spin $^c$ structures $\mathfrak {s}$ with $d(\mathfrak {s})=-k$ , where $d(\mathfrak {s})$ is the formal dimension of the Seiberg–Witten moduli space:

(5) $$ \begin{align} d(\mathfrak{s}) = \frac{1}{4}(c_1(\mathfrak{s})^2-2\chi(X)-3\sigma(X)). \end{align} $$

The group $\mathbb {Z}/2$ acts on $\mathrm {Spin}^{c}(X,k)$ by the charge conjugation, which flips the sign of the first Chern class of a spin $^c$ structure. Let $\mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ denote the quotient of $\mathrm {Spin}^{c}(X,k)$ under this $\mathbb {Z}/2$ -action. However, $\mathrm {Diff}^+(X)$ acts on $\mathrm {Spin}^{c}(X,k)$ via pull-back. Since the charge conjugation commutes with and the action of $\mathrm {Diff}^+(X)$ on $\mathrm {Spin}^{c}(X,k)$ , we have an action of $\mathrm {Diff}^+(X)$ on $\mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ .

Let $\mathcal {S}$ be a subset of $\mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ which is setwise preserved under the action of $\mathrm {Diff}^+(X)$ . We suppose that $\mathcal {S}$ does not contain the coset of a self-conjugate spin $^{c}$ structure, which is needed to ensure a families perturbation can be taken to be nonzero and transverse. We shall define a cohomology class

(6) $$ \begin{align} \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X,\mathcal{S}) \in H^k(B\mathrm{Diff}^+(X);\mathbb{Z}/2) \end{align} $$

by repeating the construction in [Reference Konno and Lin20] only for spin $^c$ structures $\mathfrak {s}$ whose cosets $[\mathfrak {s}]$ under the $\mathbb {Z}/2$ -action lie in $\mathcal {S}$ .

Before explaning the construction of $\mathbb {SW}_{\mathrm {half}\text {-}\mathrm {tot}}^k(X,\mathcal {S})$ , it is worth looking at the lowest degree case to see which spin $^c$ structures involve: suppose $k=0$ , and take a section $\tau : \mathrm {Spin}^{c}(X,0)/\mathrm {Conj} \to \mathrm {Spin}^{c}(X,k)$ of the quotient map $\mathrm {Spin}^{c}(X,0) \to \mathrm {Spin}^{c}(X,0)/\mathrm {Conj}$ . Then $\mathbb {SW}_{\mathrm {half}\text {-}\mathrm {tot}}^0(X,\mathcal {S})$ is given by

$$\begin{align*}\mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^0(X,\mathcal{S}) = \sum_{\mathfrak{s} \in \tau(\mathcal{S})}SW(X,\mathfrak{s}) \in \mathbb{Z}/2. \end{align*}$$

Note that this number in $\mathbb {Z}/2$ is independent of $\tau $ , determined only by $\mathcal {S}$ .

The characteristic class (6) is a generalization of the characteristic class (4) given in [Reference Konno and Lin20]: by setting $\mathcal {S}=\mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ , we obtain (4) – namely,

$$\begin{align*}\mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X,\mathrm{Spin}^{c}(X,k)/\mathrm{Conj}) = \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X). \end{align*}$$

3.2 Construction of the characteristic classes

We explain the construction of $\mathbb {SW}_{\mathrm {half}\text {-}\mathrm {tot}}^k(X,\mathcal {S})$ below. We omit some details which are completely analogous to arguments in [Reference Konno and Lin20]: see [Reference Konno and Lin20, Section 2] for the full treatment. First, let us recall the basics of the Seiberg–Witten equations. To write down the (perturbed) Seiberg–Witten equations, we need to fix a spin $^c$ structure $\mathfrak {s}$ on X, a Riemannian metric g on X and an imaginary-valed self-dual 2-form $\mu \in i\Omega ^+_g(X)$ on X. Here, $\Omega ^+_g(X)$ denotes the set of self-dual 2-forms for the metric g. The Seiberg–Witten equations perturbed by $\mu $ are of the form

(7) $$ \begin{align} \left\{ \begin{array}{ll} F_A^+=\sigma(\Phi, \Phi)+\mu,\\ D_A\Phi=0. \end{array} \right. \end{align} $$

Here, A is a $U(1)$ -connection of the determinant line bundle for $\mathfrak {s}$ , $\Phi $ is a positive spinor for $\mathfrak {s}$ , $\sigma (-,-)$ is a certain quadratic form, and $D_A$ is the spin $^c$ Dirac operator associated with A. The Seiberg–Witten equations is $\operatorname {\mathrm {Map}}(X,U(1))$ -equivariant, and we define the moduli space of solutions to the Seiberg–Witten equations by

$$\begin{align*}\mathcal{M}(X,\mathfrak{s},g,\mu) = \{(A,\Phi) \mid (A,\Phi) \text{ satisfies }(7)\}/\operatorname{\mathrm{Map}}(X,U(1)). \end{align*}$$

Next, let us recall the charge conjugation symmetry on the Seiberg–Witten equations. Let $\bar {\mathfrak {s}}$ denote the conjugate spin $^c$ structure to $\mathfrak {s}$ , which satisfies $c_{1}(\bar {\mathfrak {s}})=-c_{1}(\mathfrak {s})$ . Then there is a bijection

(8) $$ \begin{align} c : \mathcal{M}(X,\mathfrak{s},g,\mu) \to \mathcal{M}(X,\bar{\mathfrak{s}},g,-\mu) \end{align} $$

called the charge conjugation symmetry, which becomes a diffeomorphism between the moduli spaces if the perturbation $\mu $ is generic so that $\mathcal {M}(X,\mathfrak {s},g,\mu )$ is a smooth manifold (then so is $\mathcal {M}(X,\bar {\mathfrak {s}},g,-\mu )$ automatically).

Let $\mathscr {R}(X)$ denote the space of Riemannian metrics. Set

$$\begin{align*}\Pi(X) = \bigcup_{g \in \mathscr{R}(X)}i\Omega^+_g(X). \end{align*}$$

We think of $\Pi (X)$ as a vector bundle over the Frechet manifold $\mathscr {R}(X)$ and then take a fiberwise completion with respect to a suitable Sobolev norm. Let us use the same notation $\Pi (X) \to \mathscr {R}(X)$ also for the Hilbert bundle obtained by this completion. Let $\mathring {\Pi }(X)$ be the subset of $\Pi (X)$ consisting of perturbations $\mu $ such that:

• • $\|\mu \|\leq 1$ for the Sobolev norm on $\Omega ^+_g(X)$ , and

• • there is no reducible solution for $\mu $ .

The space $\mathring {\Pi }(X)$ is $(b^+(X)-2)$ -connected, and $\mathring {\Pi }(X)$ is invariant under the fiberwise $(-1)$ -multiplication on the Hilbert bundle $\Pi (X) \to \mathscr {R}(X)$ .

What makes the construction of the half-total Seiberg–Witten characteristic class complicated is the fact that the charge conjugation acts on the space of perturbations nontrivially; the action is given as (fiberwise) multiplication by $-1$ . Because of this, to implement a construction equivariantly under the charge conjugation, we need a ‘multi-valued perturbation’ when we form a collection of moduli spaces over a set of spin $^c$ structures, not just a copy of a common families self-dual 2-form. This is formulated as follows.

Let $\varpi : \mathrm {Spin}^{c}(X,k) \to \mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ be the quotient map. Define $\tilde {\mathcal {S}} := \varpi ^{-1}(\mathcal {S}) \subset \mathrm {Spin}^{c}(X,k)$ . Since $\mathcal {S}$ is invariant under the $\mathrm {Diff}^+(X)$ -action, $\tilde {\mathcal {S}}$ is also $\mathrm {Diff}^+(X)$ -invariant. Define

$$\begin{align*}\mathring{\Pi}(X,\mathcal{S})' := (\tilde{\mathcal{S}} \times \mathring{\Pi}(X))/(\mathbb{Z}/2), \end{align*}$$

where $\mathbb {Z}/2$ acts on $\tilde {\mathcal {S}}$ via the charge conjugation and on $\mathring {\Pi }(X)$ via the (fiberwise) $(-1)$ -multiplication. (To make our notation consistent with that in [Reference Konno and Lin20], let us use the notation $\mathring {\Pi }(X,\mathcal {S})'$ with prime, not like $\mathring {\Pi }(X,\mathcal {S})$ . This remark applies throughout this section.)

Now we consider a family of 4-manifolds. Let $X \to E \to B$ be a fiber bundle with structure group $\mathrm {Diff}^+(X)$ over a CW complex B. For $b \in B$ , we denote by $E_b$ the fiber of E over b. Associated with E, we have several natural fiber bundles. For instance, since $\mathrm {Diff}^+(X)$ acts on $\mathcal {S}$ via pull-back, we obtain an associated fiber bundle over B with fiber $\mathcal {S}$ . We denote it by

$$\begin{align*}\mathcal{S} \to \mathcal{S}(E) \to B. \end{align*}$$

Similarly, we get a fiber bundle with fiber $\mathring {\Pi }(X,\mathcal {S})'$ , denoted by

$$\begin{align*}\mathring{\Pi}(X,\mathcal{S})' \to \mathring{\Pi}(E,\mathcal{S})' \to B. \end{align*}$$

This has underlying families of spaces of metrics, denoted by

$$\begin{align*}\mathscr{R}(X) \to \mathscr{R}(E) \to B. \end{align*}$$

A section of $\mathscr {R}(E) \to B$ is a fiberwise metric on E. Note that the forgetful map $\mathring {\Pi }(X,\mathcal {S})' \to \mathcal {S}$ induces a surjection

$$\begin{align*}\mathring{\Pi}(E,\mathcal{S})' \to \mathcal{S}(E), \end{align*}$$

which commutes with the projections onto B.

It could be worth unpackaging the data $\mathring {\Pi }(E,\mathcal {S})'$ . Let $\vec {\mu }$ be a point in $\mathring {\Pi }(E,\mathcal {S})'$ . Let $b \in B$ and $g \in \mathscr {R}(E_b)$ be the images of $\vec {\mu }$ under the projections $\mathring {\Pi }(E,\mathcal {S})' \to B$ and $\mathring {\Pi }(E,\mathcal {S})' \to \mathscr {R}(E)$ . Let $\mathcal {S}(E)_b$ be the fiber of $\mathcal {S}(E) \to B$ over b. Picking a representative $\mathfrak {s}$ of each coset $[\mathfrak {s}] \in \mathcal {S}(E)_b$ , we can express $\vec {\mu }$ as a collection of a self-dual 2-forms $\{\mu _{\mathfrak {s}} \in \Omega ^+_{g}(E_b)\}_{[\mathfrak {s}] \in \mathcal {S}(E)_b}$ . We set

$$\begin{align*}\mathcal{M}(E_b, \mathcal{S}, \vec{\mu}) = \bigsqcup_{[\mathfrak{s}] \in \mathcal{S}(E)_b} \mathcal{M}(E_b, \mathfrak{s}, g, \mu_{\mathfrak{s}}). \end{align*}$$

If all $\mu _{[\mathfrak {s}]}$ are generic, $\mathcal {M}(E_b, \mathcal {S}, \vec {\mu })$ is a smooth manifold. Further, as an unoriented manifold, $\mathcal {M}(E_b, \mathcal {S}, \vec {\mu })$ is independent of choice of representatives $\mathfrak {s}$ of $[\mathfrak {s}]$ . Indeed, if we choose the other representative $\bar {\mathfrak {s}}$ of $[\mathfrak {s}]$ , the chosen perturbation becomes $\mu _{\bar {\mathfrak {s}}} = -\mu _{\mathfrak {s}}$ , so we can use the diffeomorphism (8).

Now let us take a fiberwise metric $\tilde {g} : B \to \mathscr {R}(E)$ , and pick a section $\sigma ' : \mathcal {S} \to \mathring {\Pi }(E,\mathcal {S})'$ that makes the following diagram commutative:

Define the half-total moduli space for $\sigma '$ by

$$\begin{align*}\mathcal{M}_{\mathrm{half}}(E, \mathcal{S}, \sigma') = \bigcup_{b \in B} \mathcal{M}(E_b, \mathcal{S}, \sigma'(b)). \end{align*}$$

(This was denoted by $\mathcal {M}_{\sigma ',\mathrm {half}}$ in [Reference Konno and Lin20, Definition 2.11], but let us use the notation $\mathcal {M}_{\mathrm {half}}(E,\mathcal {S}, \sigma ')$ to keep track of E and $\mathcal {S}$ .) If B is a comapct manifold, by choosing generic $\sigma '$ , $\mathcal {M}_{\mathrm {half}}(E, \mathcal {S}, \sigma ')$ becomes a compact manifold too (cf. Lemma 3.1), and the dimension of $\mathcal {M}_{\mathrm {half}}(E, \mathcal {S}, \sigma ')$ is given by $\dim {B}-k$ . In particular, for $\dim {B}=k$ , we can define a $\mathbb {Z}/2$ -valued invariant by counting the zero dimensional compact manifold $\mathcal {M}_{\mathrm {half}}(E, \mathcal {S}, \sigma ')$ .

For a general case where B is neither compact nor a manifold, we define a cochain

$$\begin{align*}\mathcal{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E, \mathcal{S}, \sigma') \in C^k(B) \end{align*}$$

as follows, where $C^k(B)$ denotes the $\mathbb {Z}/2$ -coefficient cellular cochain group. Loosely speaking, for each k-cell e of B with a characteristic map $\varphi _e : D^k \to B$ , we define

$$\begin{align*}\mathcal{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E, \mathcal{S}, \sigma')(e) = \#\mathcal{M}_{\mathrm{half}}(\varphi_e^\ast E, \mathcal{S}, \varphi_e^\ast \sigma') \in \mathbb{Z}/2. \end{align*}$$

Here, the right-hand side is a finite sum (cf. Lemma 3.1), and we can justify the necessary transversality by using a virtual neighborhood technique, just as in [Reference Konno19, Reference Konno and Lin20]. Using the assumption that $b^+(X) \geq k+2$ , we can prove that $\mathcal {SW}_{\mathrm {half}\text {-}\mathrm {tot}}^k(E, \mathcal {S}, \sigma ')$ is a cocycle, and that the cohomology class

$$\begin{align*}\mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E, \mathcal{S}) := [\mathcal{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E, \mathcal{S}, \sigma')] \in H^k(B;\mathbb{Z}/2) \end{align*}$$

is independent of the choice of $\sigma '$ ([Reference Konno and Lin20, Propositions 2.22, 2.23]). We set

$$\begin{align*}\mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X, \mathcal{S}) := \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E\mathrm{Diff}^+(X), \mathcal{S}) \in H^k(B\mathrm{Diff}^+(X);\mathbb{Z}/2). \end{align*}$$

3.3 Finiteness

Here, we record some finiteness result, which was used in Subsection 3.2 and is necessary in a subsequent argument too.

First, let us recall the following well-known finiteness of Seiberg–Witten moduli spaces (see, for example, [Reference Morgan27]). Fix a metric g and $k \in \mathbb {Z}$ . Then there are only finitely many spin $^c$ structures $\mathfrak {s}$ with $d(\mathfrak {s})=k$ for which the moduli space $\mathcal {M}(X,\mathfrak {s},g,\mu )$ for the perturbed equations (7) are nonempty for some $\mu \in \Omega ^+_g(X)$ with $\|\mu \| \leq 1$ . Here, $\|-\|$ denotes a suitable Sobolev norm. Moreover, for a fixed pair $(g,\mu )$ , the moduli space $\mathcal {M}(X,\mathfrak {s},g,\mu )$ is compact. A families generalization of this fact in our context is as follows.

As in Subsection 3.2, fix $k\geq 0$ , let X be a smooth closed oriented 4-manifold with $b^+(X) \geq k+2$ , and let $X \to E \to B$ be a fiber bundle with structure group $\mathrm {Diff}^+(X)$ over a CW complex B.

Lemma 3.1. Suppose that B is compact. If we pick a section $\sigma '$ as in Subsection 3.2, then the half-total moduli space

$$\begin{align*}\mathcal{M}_{\mathrm{half}}(E,\mathrm{Spin}^{c}(X,k)/\mathrm{Conj},\sigma') \end{align*}$$

is compact.

For our purpose, an important case is that $\mathcal {S}$ is an orbit of the action of $\mathrm {Diff}^+(X)$ on $\mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ . Set

$$\begin{align*}\mathrm{Spin}^{c}(X,k)^{\vee} := \{ \mathfrak{s} \in \mathrm{Spin}^{c}(X,k) \mid \mathfrak{s} \not\cong \bar{\mathfrak{s}} \} \end{align*}$$

and let $\mathbb {S}(X,k)$ denote the orbit space for the $\mathrm {Diff}^+(X)$ -action on $\mathrm {Spin}^{c}(X,k)^{\vee }/\mathrm {Conj}$ ,

$$\begin{align*}\mathbb{S}(X,k) = (\mathrm{Spin}^{c}(X,k)^{\vee}/\mathrm{Conj})/\mathrm{Diff}^+(X). \end{align*}$$

As an analog of the notion of a basic class, we call $\mathcal {S} \in \mathbb {S}(X,k)$ a basic orbit of E if $\mathbb {SW}_{\mathrm {half}\text {-}\mathrm {tot}}^k(E,\mathcal {S}) \neq 0$ . Let $\mathcal {B}_{\mathrm {half}}(E,k)$ denote the set of basic orbits:

$$\begin{align*}\mathcal{B}_{\mathrm{half}}(E,k) = \{\mathcal{S} \in \mathbb{S}(X,k) \mid \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E,\mathcal{S}) \neq 0\}. \end{align*}$$

Then we have the following:

4.1 Reducing to the monodromy invariant part

The characteristic class $\mathbb {SW}_{\mathrm {half}\text {-}\mathrm {tot}}^k(X, \mathcal {S})$ involves spin $^c$ structures that are not invariant under the monodromies of the families that we consider. Adapting an argument in [Reference Konno and Lin20, Section 3.1] to our setup, we shall see that such spin $^c$ structures do not contribute to the final computation.

To describe it, let us recall the numerical families Seiberg–Witten invariant. Let B be a closed smooth manifold of dimension $k \geq 0$ , X be a smooth oriented closed 4-manifold of $b^+(X) \geq k+2$ , and $X \to E \to B$ be a fiber bundle with structure group $\mathrm {Diff}^+(X)$ over B. Given a spin $^c$ structure $\mathfrak {s}$ on X of formal dimension $-k$ , suppose that the monodromy of E fixes the isomorphism class of $\mathfrak {s}$ . Then the numerical families Seiberg–Witten invariant

$$\begin{align*}SW(E,\mathfrak{s}) \in \mathbb{Z}/2 \end{align*}$$

can be defined. If the structure of E lifts to the automorphism group of the spin $^c$ 4-manifold $(X,\mathfrak {s})$ , this is the invariant defined by Li–Liu [Reference Li and Liu23]. However, even if E does not admit such a lift, one can still define $SW(E,\mathfrak {s})$ [Reference Konno19, Reference Baraglia and Konno5].

Pick an orbit $\mathcal {S} \in \mathbb {S}(X,k)$ . We regard $\mathcal {S}$ also as a subset of $\mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ . Let $\tau : \mathrm {Spin}^{c}(X,k)/\mathrm {Conj} \to \mathrm {Spin}^{c}(X,k)$ be a section of the quotient map $\mathrm {Spin}^{c}(X,k) \to \mathrm {Spin}^{c}(X,k)/\mathrm {Conj}$ . For mutually commuting diffeomorphisms $f_1, \ldots , f_k$ of X, we denote by $X_{f_1, \ldots , f_k} \to T^k$ the multiple mapping torus of $f_1, \ldots , f_k$ .

Proposition 4.1 (cf. [Reference Konno and Lin20, Corollary 3.4]).

Let $f_1, \dots , f_k : X \to X$ be mutually commuting orientation-preserving diffeomorphisms. Suppose that they satisfy the following conditions:

1. (i) For each $i = 1, \ldots , k$ , $f_i$ preserves $\tau (\mathcal {S})$ setwise.

2. (ii) For each i, there exists a smooth isotopy $(F_i^t)_{t \in [0,1]}$ from $f_i^2$ to $\mathrm {id}_X$ . For $i\neq j$ , $F_i^t$ commutes with $f_j$ for any $t \in [0,1]$ .

Then we have

$$\begin{align*}\langle \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X_{f_1, \dots, f_k}, \mathcal{S}), [T^k] \rangle = \sum_{\substack{\mathfrak{s} \in \tau(\mathcal{S}), \\ f_i^\ast \mathfrak{s} = \mathfrak{s}\ (1 \leq i \leq k)}} SW(X_{f_1, \dots, f_k},\mathfrak{s}) \end{align*}$$

in $\mathbb {Z}/2$ .

4.2 Gluing result

Another thing we need is a gluing result proven by Baraglia and the author [Reference Baraglia and Konno5]. We recall the statement for readers’ convenience. In general, let Z be an oriented smooth closed 4-manifold, and $Z \to E \to B$ be an oriented smooth fiber bundle with fiber Z. Then we get an associated vector bundle

$$\begin{align*}\mathbb{R}^{b^+(Z)} \to H^+(E) \to B \end{align*}$$

by considering maximal-dimensional positive-definite subspaces of the second cohomology fiberwise. The isomorphism class of $H^+(E)$ is determined only by E.

The gluing result we need is formulated as follows. Let $k>0$ , and let M, N be closed oriented smooth 4-manifolds with $b^+(M) \geq 2$ and $b^+(N)=k$ , and with $b_1(M)=b_1(N)=0$ . Set $X=M\#N$ . Let $\mathfrak {t} \in \mathrm {Spin}^{c}(M,0)$ and $\mathfrak {t}' \in \mathrm {Spin}^{c}(N,k+1)$ . Then we have $d(\mathfrak {t}\#\mathfrak {t}')=-k$ . Let B be a closed smooth manifold of dimension k, and $M \to E_M \to B$ and $N \to E_N \to B$ be oriented smooth fiber bundles. Fix sections $\iota _M : B \to E_M$ , $\iota _N : B \to E_N$ whose normal bundles are isomorphic via a fiberwise orientation-reversing isomorphism, so that we can form the fiberwise connected sum $X \to E_X \to B$ of $E_M$ and $E_N$ along $\iota _M, \iota _N$ . Then we have the following:

Theorem 4.2 [Reference Baraglia and Konno5, Theorem 1.1].

If $w_{b^+(N)}(H^+(E_N)) \neq 0$ , then we have

$$\begin{align*}SW(E_X,\mathfrak{t}\#\mathfrak{t}') = SW(M,\mathfrak{t}) \end{align*}$$

in $\mathbb {Z}/2$ .

Now we apply Theorem 4.2 to the multiple mapping torus $E_i \to T^k$ constructed in Subsection 2.2 for $i \geq 1$ . For each $j=1, \dots , k$ , recall that $f_j$ acts on the j-th copy of $H^+(S^2 \times S^2) \subset H^2(S^2\times S^2)$ via multiplication by $-1$ . We can see that the vector bundle

$$\begin{align*}H^+((kS^2\times S^2)_{f_1, \dots, f_k}) \to T^k \end{align*}$$

associated to the multiple mapping torus $(kS^2\times S^2)_{f_1, \dots , f_k} \to T^k$ satisfies

(9) $$ \begin{align} w_{k}(H^+((kS^2\times S^2)_{f_1, \dots, f_k})) \neq 0. \end{align} $$

Let $\mathfrak {s}_S$ denote the unique spin structure on $kS^2\times S^2$ . Then we have $\mathfrak {s}_S \in \mathrm {Spin}^{c}(kS^2\times S^2,k+1)$ .

Lemma 4.3. Let $\mathfrak {t} \in \mathrm {Spin}^{c}(M_i,0)$ . Then we have

$$\begin{align*}SW(E_i, \mathfrak{t} \# \mathfrak{s}_S) = SW(M_i,\mathfrak{t}) \end{align*}$$

in $\mathbb {Z}/2$ .

4.3 Completion of the proof

As in Section 2, fix $k>0$ , take a 4-manifold M satisfying Assumption 2.1. We shall use $M_i$ and $c_i$ that appear in Assumption 2.1, and we shall use the notation $E_i$ and $\alpha _i$ for $i\geq 1$ introduced in Subsection 2.2. Set $X = M\#kS^2 \times S^2$ and $X_i = M_i\#kS^2\times S^2$ .

For each $i\geq 1$ , we fix a section

$$\begin{align*}\tau^0_i : \mathrm{Spin}^{c}(M_i)/\mathrm{Conj} \to \mathrm{Spin}^{c}(M_i) \end{align*}$$

of the quotient map $\mathrm {Spin}^{c}(M_i) \to \mathrm {Spin}^{c}(M_i)/\mathrm {Conj}$ . Using $\tau ^0_i$ , we define a section

$$\begin{align*}\tau_i : \mathrm{Spin}^{c}(X_i)/\mathrm{Conj} \to \mathrm{Spin}^{c}(X_i) \end{align*}$$

as follows: for $\mathfrak {s} \in \mathrm {Spin}^{c}(X_i)$ , we define $\tau ([\mathfrak {s}])$ to be the spin $^c$ structure $\mathfrak {s}'$ with $[\mathfrak {s}] = [\mathfrak {s}']$ in $\mathrm {Spin}^{c}(X_i)/\mathrm {Conj}$ such that $\mathfrak {s}'|_{M_i} = \tau _0([\mathfrak {s}|_{M_i}])$ . Restricting this, we obtain a section (denoted by the same notation)

$$\begin{align*}\tau_i : \mathrm{Spin}^{c}(X_i,k)/\mathrm{Conj} \to \mathrm{Spin}^{c}(X_i,k). \end{align*}$$

As in Subsection 4.2, let $\mathfrak {s}_S$ denote the unique spin structure on $kS^2\times S^2$ . For each $i \geq 1$ , we define $\mathcal {S}_i \in \mathbb {S}(X_i,k)$ to be the $\mathrm {Diff}^+(X_i)$ -orbit that contains $[c_i\#\mathfrak {s}_S] \in \mathrm {Spin}^{c}(X_i,k)/\mathrm {Conj}$ .

Proposition 4.4. For $E_i \to T^k$ constructed in Subsection 2.2, we have

$$\begin{align*}\langle \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X, \mathcal{S}_i), (E_i)_\ast([T^k]) \rangle \neq 0 \end{align*}$$

in $\mathbb {Z}/2$ .

Proof. First, the naturality of the characteristic class implies that

(10) $$ \begin{align} \langle \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X, \mathcal{S}_i), (E_i)_\ast([T^k]) \rangle = \langle \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E_i, \mathcal{S}_i), [T^k] \rangle. \end{align} $$

To compute the right-hand side of (10), we shall apply Proposition 4.1 to the families $E_i$ . Recall that $E_i$ was constructed by using a diffeomorphism $f \in \mathrm {Diff}_\partial (S^2 \times S^2 \setminus \mathrm {Int}(D^4))$ . This diffemorphism f is order 2 in $\pi _0(\mathrm {Diff}_\partial (S^2 \times S^2 \setminus \mathrm {Int}(D^4)))$ . Thus, for the diffeomorphisms $f_1, \dots , f_k$ on $X_i$ , of which the multiples mapping torus is $E_i$ , we can find isotopies $(F_i^t)_{t \in [0,1]}$ that satisfy the assumption (ii) of Proposition 4.1. Since $f_j$ act trivially on $M_i$ , by the construction of $\tau _i$ , it follows that $\tau _i(\mathcal {S}_i)$ is setwise preserved under the actions of $f_j$ . Thus, we may apply Proposition 4.1 to the families $E_i$ and obtain the equality

(11) $$ \begin{align} \langle \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(E_i, \mathcal{S}_i), [T^k] \rangle = \sum_{\substack{\mathfrak{s} \in \tau_i(\mathcal{S}_i), \\ f_j^\ast \mathfrak{s} = \mathfrak{s}\ (1 \leq j \leq k)}} SW(E_i,\mathfrak{s}) \end{align} $$

in $\mathbb {Z}/2$ .

We shall compute the right-hand side of (11). Since $f_j$ acts on the j-th copy of $H^2(S^2\times S^2)$ via multiplication by $-1$ , a spin $^c$ structure $\mathfrak {s} \in \mathrm {Spin}^{c}(X_i)$ is $f_j$ -invariant for all j if and only if $\mathfrak {s}$ is of the form $\mathfrak {t} \# \mathfrak {s}_S$ , where $\mathfrak {t} \in \mathrm {Spin}^{c}(M_i)$ . It is easy to see that, if $d(\mathfrak {t} \# \mathfrak {s}_S)=-k$ , then $d(\mathfrak {t})=0$ . Thus, we get from Lemma 4.3 that

(12) $$ \begin{align} \sum_{\substack{\mathfrak{s} \in \tau_i(\mathcal{S}_i), \\ f_j^\ast \mathfrak{s} = \mathfrak{s}\ (1 \leq j \leq k)}} SW(E_i,\mathfrak{s}) = \sum_{\substack{\mathfrak{t}\#\mathfrak{s}_S \in \tau_i(\mathcal{S}_i), \\ \mathfrak{t} \in \mathrm{Spin}^{c}(M_i,0)}} SW(M_i,\mathfrak{t}) \end{align} $$

in $\mathbb {Z}/2$ .

To compute the right-hand side of (12), let $\mathfrak {t} \in \mathrm {Spin}^{c}(M_i,0)$ be a spin $^c$ structure on $M_i$ . We claim that $\mathfrak {t}\#\mathfrak {s}_S$ lies in $ \tau _i(\mathcal {S}_i)$ if and only if all of the following three conditions (i)–(iii) are satisfied: (i) $\mathrm {div}(c_1(\mathfrak {t})) = \mathrm {div}(c_i)$ , (ii) $c_1(\mathfrak {t})^2 = c_i^2$ , and (iii) $\mathfrak {t} \in \tau _i^0(\mathrm {Spin}^{c}(M_i,0)/\mathrm {Conj})$ . Noting $c_1(\mathfrak {t}) = c_1(\mathfrak {t} \# \mathfrak {s}_S)$ in $H^2(X_i;\mathbb {Z})$ , this claim is a direct consequence of Proposition 4.5, which we shall see later.

By the claim of the last paragraph, we have

(13) $$ \begin{align} \sum_{\substack{\mathfrak{t}\#\mathfrak{s}_S \in \tau_i(\mathcal{S}_i), \\ \mathfrak{t} \in \mathrm{Spin}^{c}(M_i,0)}} SW(M_i,\mathfrak{t}) = N(M_i; c_i) \end{align} $$

in $\mathbb {Z}/2$ . Here the right-hand side of (13) was assumed to be nonzero in $\mathbb {Z}/2$ in Assumption 2.1. Thus, the assertion of the proposition follows from (10), (11), (12), (13).

Here we record a proposition that we have used above:

Now we can complete the proof of the most general result in this paper:

Proof of Theorem 2.7.

As in the construction of $E_i$ , we fix diffeomorphisms $\psi _i : \mathring {M_i}\#kS^2\times S^2 \to \mathring {X}$ and its extensions $\psi _i : M_i\#kS^2\times S^2 \to X$ . Considering the pull-back of the orbits $\mathcal {S}_i \in \mathbb {S}(X_i,k)$ under $\psi _i$ , we obtain orbits (denoted by the same notation) $\mathcal {S}_i \in \mathbb {S}(X,k)$ .

Passing to a subsequence if necessary, we may suppose that all $\mathrm {div}(c_i)$ are distinct by (iii) of Assumption 2.1. Thus, we may suppose that all $\mathcal {S}_i$ are distinct elements in $\mathbb {S}(X,k)$ . From this together with Lemma 3.2, by passing to a subsequence again, we may suppose that

(14) $$ \begin{align} \mathcal{S}_i \notin \mathcal{B}_{\mathrm{half}}(E_1,k) \cup \cdots \cup \mathcal{B}_{\mathrm{half}}(E_{i-1},k) \end{align} $$

for all $i \geq 2$ .

Now it follows from Proposition 4.4 together with (3), (14) that the homomorphism

$$\begin{align*}\bigoplus_{i\geq2} \langle \mathbb{SW}_{\mathrm{half}\text{-}\mathrm{tot}}^k(X,\mathcal{S}_i), - \rangle : H_k(B\mathrm{Diff}^+(X);\mathbb{Z}) \to \bigoplus_{i \geq 2} \mathbb{Z}/2 \end{align*}$$

restricts to a surjection

$$\begin{align*}\left<\mathring{\alpha}_i \mid i\geq2\right> \twoheadrightarrow \bigoplus_{i \geq 2} \mathbb{Z}/2. \end{align*}$$

This combined with Lemma 2.6 implies that the subgroup $\left <\mathring {\alpha }_i \mid i\geq 2\right>$ is a $(\mathbb {Z}/2)^{\infty }$ -summand of $H_k(B\mathrm {Diff}^+(X);\mathbb {Z})$ , which together with Lemma 2.4 completes the proof of (i) of Theorem 2.7.

Since $\rho _\ast (\mathring {\alpha }_i)=\alpha _i$ , we obtain (ii) of Theorem 2.7 from (i) of Theorem 2.7 together with Lemma 2.5.

5.1 Finiteness of mapping class groups in dimension $\neq $ 4

In dimension $\neq $ 4, not only finite generation, but stronger finiteness on mapping class groups is known.

5.1.1 dimension $\geq 6$

Given a simply-connected closed smooth manifold X of $\dim X\geq 6$ , Sullivan [Reference Sullivan33, Theorem (13.3)] proved that $\pi _0(\mathrm {Diff}(X))$ is ‘commensurable’ with an arithmetic group. Krannich and Randal-Williams [Reference Krannich and Randal-Williams21] clarified that the term ‘commensurable’ is used in [Reference Sullivan33] in a different way from the current common usage. In summary, given a group, we have implications:

$$ \begin{align*} &\text{(commensurable with an arithmetic group in the current common sense)}\\ \Rightarrow& \text{(commensurable with an arithmetic group in the sense of } {[{33}])}\\ \Rightarrow & \text{(finitely presented)} \Rightarrow \text{(finitely generated)}. \end{align*} $$

In particular, Theorem 1.1 implies that mapping class groups of simply-connected 4-manifolds need not be commensurable with arithmetic groups, even in Sullivan’s sense.

5.2 Questions: finiteness in other categories

We close this paper by posting questions on categories other than the smooth category.

As noted in Remark 1.3, for a simply-connected closed topological 4-manifold X, the topological mapping class group $\pi _0(\text {Homeo}(X))$ is known to be finitely generated, and so is $H_1(B\text {Homeo}(X);\mathbb {Z})$ .

However, to the best of the author’s knowledge, there is no known finiteness result on $H_k(B\text {Homeo}(X))$ for $k>1$ for general simply-connected 4-manifolds X. However, it may be natural to hope such finiteness results in the 4-dimensional topological category, as opposed to the smooth category:

Recently, Lin and Xie [Reference Lin and Xie25] extensively studied the moduli space $\mathcal {M}^{fs}(X)$ of formally smooth 4-manifolds, which is a middle moduli space between the smooth moduli space $\mathcal {M}^{s}(X)=B\text {Diff}(X)$ and the topological moduli space $\mathcal {M}^{t}(X)=B\text {Homeo}(X)$ . Lin and Xie pointed out that most exotic phenomena detected by gauge theory are relevant to the discrepancy between $\mathcal {M}^{s}(X)$ and $\mathcal {M}^{fs}(X)$ . Since infiniteness of $\mathcal {M}^{s}(X)$ detected in this paper comes from gauge theory, it may be natural to expect finiteness of $\mathcal {M}^{fs}(X)$ :