Computing the Vasquez invariant $n(G)$ of a finite group and the sharpness of the index-sum bound for toral fibrations of compact flat manifolds

Every compact flat Riemannian manifold with finite holonomy group $G$ is a flat toral extension: it fibers over a compact flat manifold of dimension at most $n(G)$, where the Vasquez invariant $n(G)$ depends only on $G$. Yau (1972) gave the first effective bound and left its optimality open; the form of the estimate established in the later literature is due to Cliff and Weiss (1989): $n(G) \le \sum_H [G : H]$, summed over conjugacy representatives $H$ of subgroups of prime order, with equality whenever $G$ is a $p$-group, giving $n(Z_2 \times Z_2) = 6$. The surviving open problem is the exact computation of $n(G)$ for general finite groups: exact values are known for cyclic groups of squarefree order ($n = 1$, and these are the only groups with $n = 1$), for $p$-groups (the Cliff–Weiss sum; for example $n(Z_4) = 2$, $n(Z_8) = 4$, $n(Q_8) = 4$), and for isolated groups such as $A_5$ ($n = 16$, far below the index sum $62$); the groups with $n(G) = 1$ and $n(G) = 2$ are classified, but for many concrete finite groups only two-sided bounds are known, and no formula, algorithm, or structural characterization is available. The problem asks for the exact value of $n(G)$ for arbitrary finite $G$, together with the classification of the groups for which the Cliff–Weiss index-sum bound is attained (attained for every $p$-group, but not for $A_5$ and not for cyclic groups of composite squarefree order such as $Z_6$, whose bound is $5$ while $n = 1$).

Background

A compact flat Riemannian manifold is a quotient $M = \mathbb{R}^n/\Gamma$ by a torsion-free crystallographic group $\Gamma$ (a Bieberbach group). By Bieberbach's structure theory, $\Gamma$ fits into an exact sequence $0 \to L \to \Gamma \to G \to 1$ where $L \cong \mathbb{Z}^n$ is the translation lattice, the unique maximal abelian normal subgroup of $\Gamma$, and $G$ is a finite group, the holonomy group of $M$, acting faithfully on $L$ by conjugation (a $G$-lattice). Isomorphism classes of such extensions are classified by cohomology classes $\alpha \in H^2(G, L)$; the class is called special when it restricts nontrivially to every nontrivial subgroup of $G$, and special classes are exactly those giving torsion-free (Bieberbach) groups.

A flat toral extension of a compact flat manifold $N$ is a manifold of the form $(\tilde{N} \times T^k)/\pi_1(N)$, where the universal cover $\tilde{N}$ is euclidean space, $T^k = \mathbb{R}^k/\mathbb{Z}^k$ is a flat torus, and $\pi_1(N)$ acts diagonally by isometries; thus $M$ is a fiber bundle over the lower-dimensional flat manifold $N$ with flat torus fiber. Which flat manifolds are nontrivially fibered this way is classical: the Calabi construction shows that $M$ admits such a fibration exactly when its fundamental group has nontrivial center, equivalently when the first Betti number $b_1(M) = \operatorname{rank} L^G$ (the rank of the fixed sublattice) is positive. The quantitative question — how small the base can be forced to be — is not classical. Vasquez (1970) proved that for every finite group $G$ there is an integer $n(G)$ such that every compact flat manifold with holonomy $G$ is a flat toral extension of a flat manifold of dimension at most $n(G)$; algebraically, for every faithful $G$-lattice $L$ and special class $\alpha \in H^2(G, L)$ there is a $\mathbb{Z}$-pure $G$-sublattice $N \subset L$ with $\operatorname{rk}(L/N) \le n(G)$ such that the induced class $\nu^*(\alpha)$ is again special. The invariant also bounds the characteristic algebra of a flat manifold (the subalgebra of cohomology generated by its Stiefel–Whitney, Pontryagin, or Chern classes), which therefore vanishes above degree $n(G)$.

Yau (1972) gave the first effective bound (Theorem 5), together with a vanishing theorem for the characteristic algebra, and left its optimality open; his Lemma 2 gives a torus bundle over a circle for cyclic holonomy of prime order. Cliff and Weiss (1989), using permutation lattices induced from prime-order subgroups, proved the estimate $n(G) \le \sum_H [G : H]$ over conjugacy representatives $H$ of subgroups of prime order, and proved equality when $G$ is a $p$-group; in particular $n(Z_2 \times Z_2) = 6$, $n(Z_2^3) = 28$, and $n(Z_{p^k}) = p^{k-1}$, so $n(Z_4) = 2$, $n(Z_8) = 4$, $n(Z_9) = 3$, and $n(Q_8) = 4$, and the index-sum bound is sharp for the Klein four group and for every $p$-group. Cliff and Zheng (1996) computed $n(A_5) = 16$ via Scott modules, far below the index sums ($62$ for the Cliff–Weiss sum over conjugacy representatives of prime-order subgroups; $722$ if summed over the $31$ distinct maximal cyclic subgroups, which in $A_5$ all have prime order), so the bounds are not attained in general. Szczepański (1997) gave a purely algebraic definition of the invariant (coinciding with Vasquez's) and classified the groups with $n(G) = 1$: exactly the cyclic groups of squarefree order $C_{p_1 \cdots p_s}$. Filar (2014) proved subordination relations $n(H) \le n(G)$ for normal subgroups and semidirect factors and classified the groups with Vasquez number 2. Szczepański's 2006 problem list states the general question as Problem 4: calculate the Vasquez invariant $n(G)$ for a finite group $G$. Two restricted analogues indicate the difficulty: the diagonal Vasquez invariant of elementary abelian 2-groups (Chung–Petrosyan 2022) is computed exactly only up to rank 4, with exponential-versus-quadratic bounds in general, and a complex analogue for flat Kähler (generalized hyperelliptic) manifolds was introduced by Gąsior and Lutowski (2023).

Problem Statement

Let $G$ be a finite group and let $n(G)$ be the Vasquez invariant of $G$: the smallest integer such that every compact flat Riemannian manifold with holonomy group isomorphic to $G$ is a flat toral extension of a compact flat manifold of dimension at most $n(G)$. Equivalently, in algebraic terms: the smallest $k$ such that for every faithful $\mathbb{Z}G$-lattice $L$ and every special class $\alpha \in H^2(G, L)$ (a class restricting nontrivially to every nontrivial subgroup of $G$) there exists a $\mathbb{Z}$-pure $G$-sublattice $N \subset L$ with $\operatorname{rk}(L/N) \le k$ and the induced class $\nu^*(\alpha)$ special. Determine $n(G)$ exactly for arbitrary finite groups $G$. This comprises: (1) a general method — a formula, algorithm, or structural characterization — yielding $n(G)$ for any given finite group, consistent with the known values: $n(G) = 1$ exactly for the cyclic groups of squarefree order (Szczepański 1997); $n(G)$ equal to the Cliff–Weiss sum of indices of the conjugacy classes of prime-order subgroups when $G$ is a $p$-group (for example $n(Z_2 \times Z_2) = 6$, $n(Z_2^3) = 28$, $n(Z_4) = 2$, $n(Z_8) = 4$, $n(Z_9) = 3$, $n(Q_8) = 4$); $n(A_5) = 16$; and (2) the historically originating sharpness question, posed for the established form of the bound: classify the finite groups $G$ for which $n(G)$ equals the Cliff–Weiss upper bound $\sum_H [G : H]$ over conjugacy representatives $H$ of prime-order subgroups; the bound is attained for every $p$-group (Cliff–Weiss equality) and fails for $A_5$ (bound $62$ versus $n = 16$) and for cyclic groups of composite squarefree order (for example $Z_6$: bound $5$ versus $n = 1$); it is undetermined in general. This sharpness question originates in Yau's Theorem 5 and the optimality question raised there. Both parts express one objective: exact control of the minimal base dimension in the toral fibration of compact flat manifolds with prescribed holonomy.

References

1. Alphonse T. Vasquez, Flat Riemannian manifolds, Journal of Differential Geometry 4 (1970) 367–382, doi:10.4310/jdg/1214429510, https://doi.org/10.4310/jdg/1214429510
2. Shing Tung Yau, Compact flat Riemannian manifolds, Journal of Differential Geometry 6(3) (1972) 395–402, doi:10.4310/jdg/1214430500, https://doi.org/10.4310/jdg/1214430500
3. Gerald Cliff, Alfred Weiss, Torsion free space groups and permutation lattices for finite groups, Contemporary Mathematics 93, American Mathematical Society (1989) 123–132, doi:10.1090/conm/093/1003347, https://doi.org/10.1090/conm/093/1003347
4. Gerald Cliff, Hongliu Zheng, Torsion-free space groups, 2-cohomology, and Scott modules, Journal of Algebra 180(3) (1996) 889–896, doi:10.1006/jabr.1996.0100, https://doi.org/10.1006/jabr.1996.0100
5. Andrzej Szczepański, Decomposition of flat manifolds, Mathematika 44(1) (1997) 113–119, doi:10.1112/s0025579300012018, https://doi.org/10.1112/s0025579300012018
6. Andrzej Szczepański, Problems on Bieberbach groups and flat manifolds, Geometriae Dedicata 120 (2006) 111–118, arXiv:math/0507355, doi:10.1007/s10711-006-9056-1, https://arxiv.org/abs/math/0507355
7. Tomasz Filar, Flat Riemannian manifolds as torus bundles, Communications in Algebra 42(6) (2014) 2380–2387, doi:10.1080/00927872.2012.762009, https://doi.org/10.1080/00927872.2012.762009
8. Ho Yiu Chung, Nansen Petrosyan, Fibering flat manifolds of diagonal type and their fundamental groups, International Journal of Algebra and Computation 32(7) (2022) 1411–1445, arXiv:2011.07381, doi:10.1142/S0218196722500631, https://arxiv.org/abs/2011.07381
9. Anna Gąsior, Rafał Lutowski, Complex Vasquez invariant, preprint arXiv:2309.13740 (2023), https://arxiv.org/abs/2309.13740