Unboundedness of the Euler class for diffeomorphism groups of Seifert-fibered 3-manifolds (smooth-category realization of Mann's construction)

Mann (Algebr. Geom. Topol. 20 (2020) 1221-1234) proved that for a closed Seifert-fibered 3-manifold $M$ whose fiber-rotation circle includes into $\mathrm{Homeo}_0(M)$ so that $\pi_1(SO(2))\cong\mathbb{Z}$ is a direct factor of $\pi_1(\mathrm{Homeo}_0(M))$, the resulting discrete Euler class in $H^2(\mathrm{Homeo}_0(M)^\delta;\mathbb{R})$ is unbounded: for every integer $k$ there is a homomorphism from a genus-3 surface group to $\mathrm{Homeo}_0(M)$ with Euler pairing $k$, i.e. flat topological $M$-bundles over surfaces with arbitrarily large Euler class. Her construction is topological, and her Remark 3.4 states it can likely be realized in the smooth category - with the diffeomorphism group $\mathrm{Diff}_0(M)$ in place of $\mathrm{Homeo}_0(M)$ - but that more care is needed with smooth bump functions, since not all convex smooth bump functions on a disc are smoothly conjugate; no smooth realization is given. The problem posed here is that surviving open core: determine whether the Euler class in $H^2(\mathrm{Diff}_0(M)^\delta;\mathbb{R})$ is unbounded. The auxiliary question that was open when Mann's paper [1] appeared - the homotopy type of $\mathrm{Diff}$ of non-Haken infranil 3-manifolds, the last case of the generalized Smale conjecture - was resolved by Bamler-Kleiner (2021, published 2023).

Background

For a topological group $G$, write $G^\delta$ for $G$ with the discrete topology; the group cohomology $H^*(G;\mathbb{Z})=H^*(BG^\delta;\mathbb{Z})$ is the home of the characteristic classes of flat (foliated) $G$-bundles. A cohomology class is bounded if it admits a cocycle representative with finite supremum norm on tuples $(g_1,\dots,g_n)\in G^n$; a class with norm $C$ pairs with any representable surface class of a closed surface $\Sigma$ in absolute value at most $-2C\chi(\Sigma)$, which is how unboundedness is certified: a sequence of representations of a fixed surface group with unbounded pairings. The prototype of boundedness is the Milnor-Wood inequality: the Euler class of the group $\mathrm{Homeo}_0(S^1)$ of orientation-preserving homeomorphisms of the circle is bounded with norm $1/2$, so flat oriented circle bundles over $\Sigma$ have Euler number at most $-\chi(\Sigma)$ in absolute value. For a higher-dimensional fiber $M$: whenever the identity component $\mathrm{Homeo}_0(M)$ contains a circle subgroup $SO(2)\to\mathrm{Homeo}_0(M)$ (a circle action, e.g. rotation along the fibers of a fibration) whose $\pi_1(SO(2))\cong\mathbb{Z}$ injects as a direct factor of $\pi_1(\mathrm{Homeo}_0(M))$, the group $H^2(B\mathrm{Homeo}_0(M);\mathbb{Z})$ contains a corresponding $\mathbb{Z}$-summand, the Euler class of topological $M$-bundles, and one asks whether its pullback to the discrete group is bounded. Calegari proved unboundedness for $M=\mathbb{R}^2$ and Mann-Rosendal for the 2-torus. Mann's paper [1] proved the 3-dimensional Seifert-fibered case. A Seifert-fibered 3-manifold is a 3-manifold foliated by circles, i.e. a circle bundle over a 2-dimensional orbifold; rotating along the fibers gives the circle subgroup. Mann's Theorems 1.1-1.2: if $M$ is closed Seifert-fibered and the fiber-rotation inclusion makes $\pi_1(SO(2))$ a direct factor of $\pi_1(\mathrm{Homeo}_0(M))$, then every class in $H^2(\mathrm{Homeo}_0(M)^\delta;\mathbb{R})$ with nonzero restriction to the fiber circle is unbounded; indeed for each $k\in\mathbb{Z}$ there is $\rho:\pi_1(\Sigma_3)\to\mathrm{Homeo}_0(M)$ with $\langle\rho^*(e),[\Sigma_3]\rangle=k$. The proof is an explicit construction: on a fibered solid torus one builds commutators of homeomorphisms that rotate the fiber over a base point $x$ by $2\pi k f(x)$ for a standard bump function $f$; an open cover of the base orbifold by 3 sets with a subordinate partition of unity by mutually conjugate standard bumps glues three such commutators into the genus-3 surface relation, and the product of commutators of lifts to a suitable covering group $\widetilde{G}$ of $\mathrm{Homeo}_0(M)$ evaluates to exactly $k$. Which manifolds satisfy the direct-factor hypothesis is answered by the generalized Smale conjecture - computing the homotopy type of $\mathrm{Diff}(M)$, initiated by Smale's theorem that $\mathrm{Diff}(S^3)\simeq O(4)$. Hatcher and Ivanov proved that for closed Haken 3-manifolds (those containing an incompressible surface) the isometry inclusion is a homotopy equivalence, so for Seifert-fibered $M$ other than $T^3$ the fiber rotation $S^1\to\mathrm{Homeo}_0(M)$ is a homotopy equivalence (for $T^3$, $\mathrm{Homeo}_0(T^3)\simeq T^3$ with the fiber as one factor); McCullough-Soma settled the small non-Haken Seifert-fibered spaces with hyperbolic base orbifold ($\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL(2,\mathbb{R})}$ geometries, again $\mathrm{Homeo}_0(M)\simeq S^1$); Gabai settled hyperbolic $M$ (contractible identity component, so no Euler class there); the spherical cases are covered by Bamler-Kleiner and by Hong-Kalliongis-McCullough-Rubinstein. The one case still open in 2020 was the non-Haken infranil manifolds - compact quotients of the 3-dimensional Heisenberg group (Nil geometry) with no incompressible surfaces. Bamler-Kleiner resolved it: for such $X$, $\mathrm{Diff}(X)$ deformation retracts onto $\mathrm{Isom}(X,g_0)$, an extension of a finite group by $S^1$, completing the determination of the homotopy type of $\mathrm{Diff}(X)$ for all compact orientable prime 3-manifolds. Also crucial here: for 3-manifolds the inclusion $\mathrm{Diff}(M)\hookrightarrow\mathrm{Homeo}(M)$ is a homotopy equivalence (an argument of Cerf combined with Hatcher's proof of the Smale conjecture, recounted in [1]), so homotopy-type statements transfer between the categories. Boundedness, however, is a property of the discrete group, not of the homotopy type. Thurston's theorem that $B\mathrm{Homeo}(M)^\delta\to B\mathrm{Homeo}(M)$ induces an isomorphism on homology underlies the topological picture but has no analogue for diffeomorphism groups, whose discrete cohomology is richer (e.g. the Godbillon-Vey class of $\mathrm{Diff}_0(S^1)$ is a regularity-sensitive phenomenon). By Ivanov's theorem the fiber rotation $S^1\to\mathrm{Diff}_0(M)$ is a homotopy equivalence for Haken Seifert-fibered $M$, so the Euler class $e\in H^2(B\mathrm{Diff}_0(M);\mathbb{Z})$ exists, restricts to a generator on the fiber circle, and one may ask whether its pullback $e^\delta\in H^2(\mathrm{Diff}_0(M)^\delta;\mathbb{R})$ is bounded. Mann's Remark 3.4 [1] expects the smooth answer to match the topological one but leaves the realization open, precisely because the gluing step uses that all standard topological bump functions on a disc are conjugate by a homeomorphism, whereas smooth conjugacy must preserve the differential geometry of the level sets of the bump, so the local commutator lemma has no direct smooth analogue.

Problem Statement

Let $M$ be a closed, orientable, Seifert-fibered 3-manifold such that the fiber-rotation inclusion $SO(2)\hookrightarrow\mathrm{Diff}_0(M)$ induces $\pi_1(SO(2))\cong\mathbb{Z}$ as a direct factor of $\pi_1(\mathrm{Diff}_0(M))$ - the smooth analogue of the hypothesis of Mann's Theorem 1.1. The hypothesis holds, for example, for every closed Haken Seifert-fibered $M$ (for $M\neq T^3$ the fiber rotation $S^1\to\mathrm{Diff}_0(M)$ is a homotopy equivalence by Ivanov; for $T^3$ one has $\mathrm{Diff}_0(T^3)\simeq T^3$ with the fiber rotation as one circle factor), and, by Bamler-Kleiner's completion of the generalized Smale conjecture, for non-Haken infranil manifolds, where $\mathrm{Diff}(X)\simeq\mathrm{Isom}(X)$ is an extension of a finite group by the fiber-rotation circle. Let $e\in H^2(B\mathrm{Diff}_0(M);\mathbb{Z})$ be the corresponding Euler class - the class of the central extension $0\to\mathbb{Z}\to\widetilde{G}\to\mathrm{Diff}_0(M)\to 1$, where $\widetilde{G}$ is the covering group of $\mathrm{Diff}_0(M)$ corresponding to the complementary factor of $\pi_1(SO(2))$ in $\pi_1(\mathrm{Diff}_0(M))$ - and let $e^\delta\in H^2(\mathrm{Diff}_0(M)^\delta;\mathbb{R})$ be its pullback to the discrete group. Equivalently, for a representation $\rho:\pi_1(\Sigma)\to\mathrm{Diff}_0(M)$ of a closed oriented surface group, $\langle\rho^*(e^\delta),[\Sigma]\rangle$ is the obstruction to lifting $\rho$ to $\widetilde{G}$, computable as the total exponent of the product of commutators of lifts of the standard generators. Question: is $e^\delta$ unbounded? Concretely: does there exist, for every integer $k$, a homomorphism $\rho:\pi_1(\Sigma_3)\to\mathrm{Diff}_0(M)$ with $\langle\rho^*(e),[\Sigma_3]\rangle=k$ - i.e. flat smooth $M$-bundles over a fixed genus-3 surface with arbitrarily large Euler class? Mann's Theorem 1.2 gives exactly this with $\mathrm{Diff}_0$ replaced by $\mathrm{Homeo}_0$; establishing the smooth version resolves the problem in the expected direction. A resolution in the opposite direction - a smooth Milnor-Wood inequality $|\langle\rho^*(e),[\Sigma]\rangle|\le C\cdot|\chi(\Sigma)|$ valid for all homomorphisms $\rho:\pi_1(\Sigma)\to\mathrm{Diff}_0(M)$ - equally resolves it. The primary case of interest is that $S^1\to\mathrm{Diff}_0(M)$ be a homotopy equivalence, e.g. any closed Haken Seifert-fibered $M\neq T^3$.

References

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