On Torelli groups and Dehn twists of smooth 4-manifolds

Recently, Gay [Gay21, Theorem 1] constructed a surjection

onto the smooth oriented mapping class group of $S^{4}$, whose domain is a colimit of the fundamental groups of the space of smooth embeddings of $\sqcup^{g}S^{2}$ into $W_{g,1}\coloneq(S^{2}\times S^{2})^{\sharp g}\setminus{\mathrm{int}}(D^{4})$, based at the embedding $\mathrm{inc}\colon\sqcup^{g}S^{2}\hookrightarrow W_{g,1}$ induced by $g$ copies of the inclusion $S^{2}\times\{\ast\}\subset S^{2}\times S^{2}$. In the first part of this note, we use work of Kreck [Kre79] and Quinn [Qui86] to prove a generalisation of Gay’s result which applies to all simply connected closed oriented smooth 4-manifolds $X$. For such a general $X$, the target of (1) needs to be replaced by the Torelli subgroup ${\mathrm{Tor}}(X)\subset\mathrm{Diff}^{+}(X)$—the subgroup of those diffeomorphisms that act as the identity on the integral homology of $X$.

Related to, but independent of, the proof of A (see 3), we show in the second part of this note that the Dehn twist $t_{X}\in\pi_{0}(\mathrm{Diff}_{\partial}(X\sharp D^{4}))$ along the boundary sphere of $X\sharp D^{4}$ for a closed smooth simply-connected $4$-manifold is stably isotopic to the identity. In addition to Kreck’s and Quinn’s work mentioned above, this relies on Wall’s stable classification in [Wal64] and Galatius–Randal-Williams’ work on stable moduli spaces of manifolds [GRW17].

We now turn to the proofs of Theorem A and B, in this order.

Consider the embedding $\smash{\sqcup^{g}D^{2}\times S^{2}\subset W_{g,1}=(S^{2}\times S^{2})^{\sharp g}\backslash\mathrm{int}(D^{4})}$ obtained by viewing $D^{2}\subset S^{2}$ as the upper hemisphere, performing the connected sums in the complement of $\sqcup^{g}D^{2}\times S^{2}\subset\sqcup^{g}S^{2}\times S^{2}$, and choosing the embedded $4$-disc also away from this complement. Note that the complement $Y_{g,1}\coloneq W_{g,1}\backslash\sqcup^{g}{\mathrm{int}}(D^{2}\times S^{2})$ has the diffeomorphism type of the connected sum $D^{4}\sharp\smash{(D^{2}\times S^{2})^{\sharp g}}$.

Recall that the Dehn twist $t_{X}$ is the image of the non-trivial element under the leftmost map in (4). To study its behaviour under stabilisation, we use that the Dehn twists in $\pi_{0}(\mathrm{Diff}_{\partial}(W_{1,1}\sharp D^{4}))$ around the two boundary components are isotopic (see [Gia08, Proposition 3.7]), so the stabilisation map $\smash{\pi_{0}(\mathrm{Diff}_{\partial}(X\sharp W_{g,1}))\to\pi_{0}(\mathrm{Diff}_{\partial}(X\sharp W_{g+1,1}))}$ sends $t_{X\sharp W_{g}}$ to $t_{X\sharp W_{g+1}}$. Combined with (4) and writing $W_{g}\coloneq\sharp^{g}S^{2}\times S^{2}$ this shows that there is a stabilisation map $\pi_{0}(\mathrm{Diff}^{+}(X\sharp W_{g}))\to\pi_{0}(\mathrm{Diff}^{+}(X\sharp W_{g+1}))$, a well-defined element $t_{X\sharp W_{\infty}}\in\mathrm{colim}_{g\to\infty}\,\pi_{0}(\mathrm{Diff}_{\partial}(X\sharp W_{g,1}))$, an exact sequence

and that B is equivalent to showing that $t_{X\sharp W_{\infty}}$ is trivial. Since every orientation-preserving diffeomorphism between oriented manifolds can be assumed to fix a codimension $0$ disc, this question depends only on the oriented diffeomorphism type of $X$ up to taking connected sums with $S^{2}\times S^{2}$, the stable diffeomorphism type. The latter is by [Wal64, Theorem 2 and 3] determined by the stable isomorphism type of the intersection form, i.e. the isomorphism type of the intersection form up to adding copies of the hyperbolic form $H\coloneq(\mathbf{Z}^{2},\left[\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right])$, the intersection form of $S^{2}\times S^{2}$.

If $X$ is not spin, then the Dehn twist $t_{X}$ is trivial even before stabilisation (see the remark in the introduction). If $X$ is spin, then its intersection form is even and its signature is divisible by $16$ as a consequence of Rokhlin’s theorem. From the classification of indefinite unimodular symmetric bilinear forms over $\mathbf{Z}$ [MH73, Theorem II.5.3], it follows that the stable isomorphism type (i.e. isomorphism up to addition of $H$) of the intersection form of $X$ agrees with that of $\smash{(E_{8}^{\oplus-2}\oplus H^{\oplus 3})^{\oplus k}}$ for some $k\in\mathbf{Z}$, which is the intersection form of $\smash{\sharp^{k}K_{3}}$ (the convention is that $\smash{\sharp^{k}K_{3}}$ is for negative $k$ the $|k|$-fold connected sum of $K_{3}$ with the opposite orientation). We may thus assume $\smash{X=\sharp^{k}K_{3}}$ for some $k\in\mathbf{Z}$. To settle this case, we first prove the following result on stable abelianisations.

In view of 4, to show that $t_{X\sharp W_{\infty}}$ is trivial, it suffices to prove that the groups $\mathrm{H}_{1}(\pi_{0}(\mathrm{Diff}_{\partial}(W_{\infty,1})))$ and $\mathrm{H}_{1}(\pi_{0}(\mathrm{Diff}_{\partial}(X\sharp W_{\infty,1})))$ are abstractly isomorphic. As $\mathrm{H}_{1}(\pi_{0}(G))\cong\mathrm{H}_{1}(\mathrm{B}G)$ holds for any topological group $G$ (because $\pi_{1}(\mathrm{B}G)\cong\pi_{0}(G)$ and $\mathrm{H}_{1}(A)$ agrees with the abelianisation of $\pi_{1}(A)$ for any connected space $A$), this is equivalent to showing that the first homology groups of the homotopy colimits

are isomorphic in the two cases $Z=S^{4}$ and $Z=X=\sharp^{k}K_{3}$. Here we used that taking homology turns sequential homotopy colimits into colimits (for a reference, use the mapping telescope model for homotopy colimits and apply [May99, Section 14.6]). We will show the more general statement that the homology groups of the spaces (8) are in any degree independent of the choice of $Z$ up to isomorphism for any closed $1$-connected spin manifold $Z$, in particular for $Z=S^{4}$ and $Z=X=\sharp^{k}K_{3}$. This independence result follows from work of Galatius–Randal-Williams: an application of [GRW17, Theorem 1.5] to the map $\theta\colon\mathrm{BSpin}(4)\rightarrow\mathrm{BO}(4)$ shows that there is for any $1$-connected closed spin manifold $Z$ a map from (8) to a fixed infinite loop space $\Omega^{\infty}\mathrm{MT\theta}$, independent of $Z$, that is a homology equivalence onto the path component of $\Omega^{\infty}\mathrm{MT\theta}$ that is hit by this map; which path component may depend on $Z$. Deducing this from the cited result uses that, by an application of obstruction theory, the spaces $\mathrm{BDiff}_{\partial}(W)$ and $\mathrm{Bun}_{n,\partial}^{\theta}(TW;\hat{\ell}_{P})\sslash\mathrm{Diff}_{\partial}(W)$ in Definition 1.1 loc.cit. are homotopy equivalent for $n=2$, $P=S^{3}$, and $W=Z\sharp W_{g,1}$. Since all path components of a loop space are homotopy equivalent to the component of the constant loop (by multiplication with the inverse loop), all path components of $\Omega^{\infty}\mathrm{MT\theta}$ have the same homology groups, so the claim follows.

We thank David Gay for a conversation regarding the first part of this note, Peter Kronheimer for answering some questions related to [KM20], Mark Powell for pointing us to [Sae06], and the anonymous referee for their comments and suggestions.