Cohomology of $B\mathrm{Diff}$ for connected sums of lens spaces beyond the generic non-diffeomorphic case

For a smooth 3-manifold $M$, the rational cohomology of $B\mathrm{Diff}(M)$, the classifying space of the diffeomorphism group, is the ring of rational characteristic classes of smooth $M$-bundles. Such computations long remained confined to prime 3-manifolds, where the homotopy type of $\mathrm{Diff}(M)$ is known through Smale-conjecture-type results. Lelkes (2024) computed the first complete example for a non-prime manifold whose diffeomorphism group is not of compact-Lie type — the connected sum $L_1\#L_2$ of two non-diffeomorphic generic lens spaces — and the prime decomposition fibre sequence of Boyd–Bregman–Steinebrunner (2026) now gives the general framework for reducible 3-manifolds. The open problem is to determine $H^*(B\mathrm{Diff}(M);\mathbb{Q})$ for connected sums of lens spaces outside the computed case — two copies of the same lens space, sums involving non-generic lens spaces, and three or more summands — and to identify the homotopy type of $\mathrm{Diff}(M)$ by an explicit model. Two clarifications are in order: the flagship case $L(p,q)\#L(p,q')$ (generic, non-diffeomorphic) is already computed, and the non-prime manifold $\mathbb{R}P^3\#\mathbb{R}P^3$ has a known diffeomorphism-group homotopy type.

Background

For a smooth compact manifold $M$, let $\mathrm{Diff}(M)$ be its group of diffeomorphisms with the $C^\infty$-topology. The classifying space $B\mathrm{Diff}(M)$ parametrizes smooth fibre bundles with fibre $M$: concordance classes of smooth $M$-bundles over a base $X$ correspond to homotopy classes of maps $X\to B\mathrm{Diff}(M)$, so the cohomology of $B\mathrm{Diff}(M)$ is exactly the theory of characteristic classes of smooth $M$-bundles. In dimension two this cohomology is described in a stable range by the theorem on the stable cohomology of moduli spaces of surfaces (the Mumford conjecture), and analogues exist in high even dimensions; in dimension three, complete computations of $H^*(B\mathrm{Diff}(M);\mathbb{Q})$ were, until very recently, available essentially only when the homotopy type of $\mathrm{Diff}(M)$ itself was known.

Every compact orientable 3-manifold admits an essentially unique prime decomposition $M=P_1\#\cdots\#P_n\#(S^1\times S^2)^{\#g}$ into prime summands, where $\#$ denotes connected sum (removing a 3-ball from each of two manifolds and gluing along the boundary spheres) and the prime summands other than $S^1\times S^2$ are irreducible, meaning every embedded 2-sphere in them bounds a 3-ball. A 3-manifold is called reducible (non-prime) if it contains an embedded 2-sphere that bounds no 3-ball, i.e. if it is a nontrivial connected sum. Lens spaces $L(m,q)$, with $m,q$ coprime, are the quotients of $S^3\subset\mathbb{C}^2$ by the cyclic action $(z_1,z_2)\mapsto(e^{2\pi i/m}z_1,e^{2\pi iq/m}z_2)$; they are irreducible, and their diffeomorphism groups are completely understood: the generalized Smale conjecture — the statement that the inclusion of the isometry group $\mathrm{Isom}(M)\hookrightarrow\mathrm{Diff}(M)$ is a homotopy equivalence, generalizing Hatcher's theorem $\mathrm{Diff}(S^3)\simeq O(4)$ — holds for all elliptic 3-manifolds (quotients of the round 3-sphere by a finite free action of isometries, also called spherical space forms), hence for all lens spaces (Hong–Kalliongis–McCullough–Rubinstein), for all spherical space forms independently (Bamler–Kleiner, via Ricci flow), and for closed hyperbolic 3-manifolds (closed manifolds carrying a complete metric of constant negative sectional curvature; Gabai). For such prime $M$, $H^*(B\mathrm{Diff}(M);\mathbb{Q})$ is the cohomology of the classifying space of a compact Lie group (a finite group in the hyperbolic case).

For reducible $M$ the picture is different. Hatcher computed $\mathrm{Diff}(S^1\times S^2)\simeq O(2)\times O(3)\times\Omega O(3)$, where $\Omega O(3)$ is the space of based loops in the orthogonal group $O(3)$ — the first diffeomorphism group not homotopy equivalent to a Lie group — and his method handles both $S^2$-bundles over $S^1$, the product and the twisted one. For a connected sum of two irreducible 3-manifolds, a theorem of Hatcher gives a deformation retraction of $\mathrm{Diff}(M)$ onto the subgroup preserving a splitting 2-sphere, which essentially reduces the homotopy type to the summands. A programme announced by César de Sá and Rourke and completed by Hendriks–Laudenbach and Hendriks–McCullough expressed $\mathrm{Diff}^+_\partial(M)$ (the group of orientation-preserving diffeomorphisms fixing the boundary pointwise) through the diffeomorphism groups of the punctured prime factors via a homotopy fibre sequence, and Hatcher proposed in an unfinished draft a delooping of that sequence to the level of classifying spaces. This delooping has now been carried out: Boyd–Bregman–Steinebrunner constructed the prime decomposition fibre sequence $\mathcal{H}_g(P_1,\dots,P_n)\longrightarrow B\mathrm{Diff}^+(M)\longrightarrow B\mathrm{Diff}^+(P_1\sqcup\cdots\sqcup P_n),$ where $\mathrm{Diff}^+$ denotes the orientation-preserving diffeomorphisms and the subscript $g$ is the number of $S^1\times S^2$ summands in the prime decomposition of $M$ above; the fibre $\mathcal{H}_g$ is a moduli space of 1-handle attachments (recording where the connecting tubes of the connected sum are attached), describable as a finite homotopy colimit of configuration spaces on the prime pieces and, for $n>0$, equivalent to a finite cell complex; the same authors proved that $B\mathrm{Diff}(M)$ has finite homotopy type for every compact orientable 3-manifold $M$ (Kontsevich's finiteness conjecture, with boundary, appears as Problem 3.48 on the Kirby problem list). For a connected sum $M=L_1\#\cdots\#L_n$ of lens spaces, which has no $S^1\times S^2$ summand, this sequence specializes to its $g=0$ case $\mathcal{H}_0(L_1,\dots,L_n)\longrightarrow B\mathrm{Diff}^+(M)\longrightarrow B\mathrm{Diff}^+(L_1\sqcup\cdots\sqcup L_n),$ whose base is identified through the generalized Smale conjecture, whose fibre is a homotopy colimit of configuration spaces on the summands, and on which the remaining components of $\mathrm{Diff}(M)$ act.

Against this backdrop, Lelkes computed the first complete rational cohomology ring of $B\mathrm{Diff}$ for a non-prime 3-manifold whose diffeomorphism group is not of compact-Lie type. For $M=L_1\#L_2$, the connected sum of two non-diffeomorphic generic lens spaces (generic meaning $m>2$, $1<q<m/2$, and $q^2\not\equiv\pm1\pmod m$; such lens spaces admit no orientation-reversing diffeomorphism), he obtained $H^*(B\mathrm{Diff}(L_1\#L_2);\mathbb{Q})\cong\mathbb{Q}[\mu^2,\eta^2,\nu^2,\vartheta^2]/(\mu^2\eta^2,\,\nu^2\vartheta^2,\,\mu^2+\eta^2-\nu^2-\vartheta^2),$ and computed the mapping class group — the group of isotopy classes of diffeomorphisms, $\pi_0\mathrm{Diff}(M)$ — finding $\pi_0\mathrm{Diff}(L_1\#L_2)\cong C_2\times C_2$ ($C_2$ the cyclic group of order two), and proved that $B\mathrm{Diff}(L_1\#L_2)$ is not weakly equivalent to $BG$ for any compact Lie group $G$ — so a literal Smale-type identification of $\mathrm{Diff}$ by a compact Lie group is impossible in general for reducible $M$, and the fibre-sequence description is the correct form of answer. Boyd–Bregman–Steinebrunner computed the analogous ring for $(S^1\times S^2)^{\#2}$ and described $B\mathrm{Diff}_0(M)$ completely when the irreducible factors are aspherical (i.e. have contractible universal cover). Lens spaces are not aspherical, and beyond Lelkes's generic non-diffeomorphic case and the single further settled sum $\mathbb{R}P^3\#\mathbb{R}P^3$ (whose diffeomorphism group is known to be the compact Lie group $\mathrm{O}(1)\times\mathrm{O}(3)$) no computation of $H^*(B\mathrm{Diff}(M);\mathbb{Q})$ exists for any other connected sum of lens spaces.

Problem Statement

Let $M$ be a connected sum of 3-dimensional lens spaces, and let $\mathrm{Diff}(M)$ be its full diffeomorphism group with the $C^\infty$-topology. Determine, for every such $M$, the rational cohomology ring $H^*(B\mathrm{Diff}(M);\mathbb{Q})$ — equivalently, all rational characteristic classes of smooth $M$-bundles. Since $B\mathrm{Diff}(L_1\#L_2)$ is known not to be weakly equivalent to the classifying space of any compact Lie group, compact-Lie answers are not available in general.

The case of two non-diffeomorphic generic lens spaces is settled (Lelkes) and excluded, as is the single-summand case of a lone lens space, which is prime and settled by the generalized Smale conjecture (see Background). The open core, whose three regimes jointly exhaust the remaining connected sums of lens spaces and share the single objective above, is:

1. Diffeomorphic summands. $M=L(p,q)\#L(p,q)$ with $L(p,q)$ generic. Compute $H^*(B\mathrm{Diff}(M);\mathbb{Q})$, treating the order-two diffeomorphism permuting the two summands, the resulting larger mapping class group (the group of isotopy classes of diffeomorphisms, $\pi_0\mathrm{Diff}(M)$), and its action on $H^*(B\mathrm{Diff}_0(M);\mathbb{Q})$.

2. Non-generic summands. $M=L_1\#L_2$ where at least one summand fails the genericity condition, i.e. $q^2\equiv\pm1\pmod m$ or the excluded parameter ranges (including $L(2,1)=\mathbb{R}P^3$). Such summands may admit orientation-reversing diffeomorphisms and larger isometry groups, changing $\pi_0$ of the pieces and the gluing analysis. Compute $H^*(B\mathrm{Diff}(M);\mathbb{Q})$, either for the full group or for $\mathrm{Diff}^+$ together with the action of the orientation-reversing components. (The single settled point $L(2,1)\#L(2,1)=\mathbb{R}P^3\#\mathbb{R}P^3$, where $\mathrm{Diff}(M)\simeq\mathrm{O}(1)\times\mathrm{O}(3)$ by Bamler–Kleiner's Theorem 1.5 and hence $H^*(B\mathrm{Diff};\mathbb{Q})$ is known, is excluded; all other non-generic parameter choices remain open.)

3. Three or more summands. $M=L_1\#\cdots\#L_n$ with $n\ge3$, the $L_i$ arbitrary (possibly repeated and non-generic). Compute $H^*(B\mathrm{Diff}(M);\mathbb{Q})$; here the fibre $\mathcal{H}_0(L_1,\dots,L_n)$ is a genuine homotopy colimit over dual graphs of separating-sphere systems, so the computation involves configuration spaces on the summands, possible higher derived-limit terms, and the action of the finite mapping class group $\pi_0\mathrm{Diff}(M)$.

References

1. Zoltán Lelkes, Cohomology of the diffeomorphism group of the connected sum of two generic lens spaces, 2024, arXiv:2412.11225, https://arxiv.org/abs/2412.11225
2. Rachael Boyd, Corey Bregman, and Jan Steinebrunner, The prime decomposition fibre sequence for moduli spaces of reducible 3-manifolds, 2026, arXiv:2604.02099, https://arxiv.org/abs/2604.02099
3. Rachael Boyd, Corey Bregman, and Jan Steinebrunner, Moduli spaces of 3-manifolds with boundary are finite, 2024, arXiv:2404.12748, https://arxiv.org/abs/2404.12748
4. Allen E. Hatcher, A proof of the Smale conjecture, Diff(S^3) ≃ O(4), Annals of Mathematics (2) 117 (1983), no. 3, 553–607, DOI:10.2307/2007035, https://doi.org/10.2307/2007035
5. Allen Hatcher, On the diffeomorphism group of S^1 × S^2, Proceedings of the American Mathematical Society 83 (1981), no. 2, 427–430, DOI:10.1090/S0002-9939-1981-0624946-2, https://doi.org/10.1090/S0002-9939-1981-0624946-2; revised December 2003, https://pi.math.cornell.edu/~hatcher/Papers/newDiffS1xS2.pdf
6. Allen Hatcher, Diffeomorphism groups of reducible 3-manifolds, unfinished draft, https://pi.math.cornell.edu/~hatcher/Papers/DR3M.pdf
7. Sungbok Hong, John Kalliongis, Darryl McCullough, and J. Hyam Rubinstein, Diffeomorphisms of Elliptic 3-Manifolds, Lecture Notes in Mathematics 2055, Springer, 2012, DOI:10.1007/978-3-642-31564-0, arXiv:1110.4996, https://arxiv.org/abs/1110.4996
8. Richard H. Bamler and Bruce Kleiner, Ricci flow and contractibility of spaces of metrics, 2019, arXiv:1909.08710, https://arxiv.org/abs/1909.08710
9. David Gabai, The Smale conjecture for hyperbolic 3-manifolds: Isom(M^3) ≃ Diff(M^3), Journal of Differential Geometry 58 (2001), no. 1, 113–149, DOI:10.4310/jdg/1090348284, https://doi.org/10.4310/jdg/1090348284
10. Eugénia César de Sá and Colin Rourke, The homotopy type of homeomorphisms of 3-manifolds, Bulletin of the American Mathematical Society (N.S.) 1 (1979), no. 1, 251–254, DOI:10.1090/S0273-0979-1979-14574-9, https://doi.org/10.1090/S0273-0979-1979-14574-9
11. Harrie Hendriks and François Laudenbach, Difféomorphismes des sommes connexes en dimension trois, Topology 23 (1984), no. 4, 423–443, DOI:10.1016/0040-9383(84)90004-1, https://doi.org/10.1016/0040-9383(84)90004-1
12. Harrie Hendriks and Darryl McCullough, On the diffeomorphism group of a reducible 3-manifold, Topology and its Applications 26 (1987), no. 1, 25–31, DOI:10.1016/0166-8641(87)90023-X, https://doi.org/10.1016/0166-8641(87)90023-X
13. Darryl McCullough, Isometries of elliptic 3-manifolds, 2000, arXiv:math/0010077, https://arxiv.org/abs/math/0010077; published in the Journal of the London Mathematical Society (2) 65 (2002), DOI:10.1112/S0024610701002782, https://doi.org/10.1112/S0024610701002782
14. Sam Nariman, On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds, 2021, arXiv:2104.12338, https://arxiv.org/abs/2104.12338; published in Forum of Mathematics, Sigma 13 (2025), DOI:10.1017/fms.2025.38, https://doi.org/10.1017/fms.2025.38
15. Riccardo Benedetti and Paolo Lisca, Framing 3-manifolds with bare hands, 2018, arXiv:1806.04991, https://arxiv.org/abs/1806.04991