ON FUNDAMENTAL GROUPS OF POSITIVELY CURVED MANIFOLDS

KRISHNAN SHANKAR

0. Introduction

In 1965, S.S.Chern posed the following question [7, p.167] (sometimes called Chern’s conjecture [12, p.671]; see also [6]): Let M be a compact Riemannian manifold of positive sectional curvature. Is it true that every abelian subgroup of π_{1}(M) is cyclic? Since π_{1}(M) is finite, this is equivalent to saying that the cohomology ring H^{∗}(π_{1}, ℤ) is periodic (cf. [2]). In this note we will point out that there exist infinitely many counterexamples by observing that the normal homogeneous Aloff-Wallach space N_{1,1} (cf. [10]) and the Eschenburg space M_{1,1} (cf. [3])^{1} both admit free, isometric SO(3) actions. Curiously enough N_{1,1} was precisely the one missed in the classification of positively curved normal homogeneous spaces (cf. [1]). So, the motivation for posing the question possibly came from looking at metric space forms (cf. [11]) or more generally (?) \frac{1}{4}-pinched manifolds (cf. [4]) where the fundamental groups all have periodic cohomology, and from manifolds of negative curvature where the statement is true (Preissman’s Theorem).

1. Free, isometric SO(3) actions

Following Wilking [10] we represent the normal homogeneous Aloff-Wallach space N_{1,1} as the quotient (SU(3) × SO(3))/ U^{∗}(2). Here U^{∗}(2) is the image under the embedding (i, π) : U(2) ↪ SU(3) × SO(3) given by the natural inclusion

i(A) = (\begin{matrix}A & 0 \\ 0 & det(A)^{−1} \end{matrix}) for A ∈ U(2)

and the projection π : U(2) → U(2)/S^{1} ≅ SO(3), where S^{1} ⊂ U(2) is the center of U(2). The metric being normal homogeneous, the entire group SU(3) × SO(3) acts isometrically on N_{1,1} on the left. In particular, the subgroup {id} × SO(3) acts isometrically on the left.

Proposition 1.1. The group {id} × SO(3) acts freely on N_{1,1}.

Proof: The action is free if and only if ({id}×SO(3))∩Ad(g)(U^{∗}(2)) is trivial for all g in SU(3) × SO(3). This is equivalent to saying that Ad(g)({id} × SO(3)) ∩ U^{∗}(2) is trivial for all g. But Ad(g)({id} × SO(3)) = {id} × SO(3) and (i, π)(U(2)) ∩ {id} × SO(3) is clearly trivial. □

1In [3], the space M_{1,1} is denoted as M′_{1,1}.

The Eschenburg space M_{1,1} is constructed as follows: Start with the group U(3) and perturb the bi-invariant metric to a normal homogeneous metric that is left invariant and Ad(U(2) × U(1))-invariant. Consider the subgroups Z′ = {diag(z, z, z̄)|z ∈ S^{1}} and U_{p,q} = {diag(z^{p}, z^{q}, 1)|z ∈ S^{1}} where gcd(p, q) = 1 and diag(a_{1}, a_{2}, …, a_{n}) denotes the matrix with diagonal entries a_{1}, a_{2}, …, a_{n}. It is shown in [3] that if p · q > 0 then the double coset manifold M_{p,q} = U_{p,q}\ U(3)/Z′ (also called a biquotient) has positive curvature for the submersed metric. Since the group U(2) × Z′ acts freely and isometrically on U(3), it follows that there is a Riemannian fibration (see [3] for details)

(U(2) × Z′)/(U_{p,q} × Z′) → M_{p,q} → CP^{2}

When p = q = 1, U(2) × Z′ induces an isometric but non-effective action on M_{1,1} (since U_{1,1} is the center of U(2)) with kernel U_{1,1} × Z′. The resulting isometric action by SO(3) = (U(2) × Z′)/(U_{1,1} × Z′) is clearly free and we get

SO(3) → M_{1,1} → CP^{2}

Proposition 1.2. The Eschenburg space M_{1,1} admits a free, isometric SO(3) action. □

In fact, it was consideration of this fibration that led to the original observation by the author.

2. Remarks

1. Up to conjugacy the finite subgroups of SO(3) are

ℤ_{n}, n ≥ 1  D_{m}, m ≥ 2  A_{4}  S_{4}  A_{5}

where D_{m} is the dihedral group of order 2m, S_{n} denotes the permutation group on n letters, and A_{n} ⊂ S_{n} is the subgroup of even permutations. Since ℤ_{2} ⊕ ℤ_{2} ≈ D_{2} ↪ D_{2m} for all m ≥ 1, we get two infinite families of counterexamples: one covered by N_{1,1} and one covered by M_{1,1}. This answers Chern’s question in the negative.

2. A finite group is said to satisfy the m-condition if any subgroup of order m is cyclic. It is shown in [5] that a finite group acts freely on a topological sphere if and only if it satisfies all 2p- and p^{2}-conditions where p is any prime that divides the order of the group. Note that satisfying all p^{2}-conditions is equivalent to the condition in Chern’s problem. The above examples show that neither the 2p- nor the 2^{2}-conditions need hold for fundamental groups of positively curved manifolds. However, it is not known whether the p^{2}-condition remains true for odd primes p. We may formulate the following:

Question. Let M be a compact Riemannian manifold of positive sectional curvature. Is it true that every abelian subgroup of π_{1}(M) of odd order is cyclic?

3. In the context of the question, some partial answers are known when the dimension of the manifold is fixed; see for instance [8].

4. The remaining proper, closed subgroups of SO(3) are SO(2) ≈ S^{1} and O(2). The quotients by SO(2) are N_{1,1}/S^{1} = F and M_{1,1}/S^{1} = F′ where F is the space of flags over CP^{2} and F′ is the “twisted” Eschenburg flag (cf. [3]). The quotients by O(2) then give positively curved manifolds with fundamental group ℤ_{2}. They are isometric ℤ_{2} quotients of F and F′ respectively. It follows from Synge’s theorem that they are nonorientable. It can be shown without too much difficulty that the quaternionic flag Sp(3)/(Sp(1) × Sp(1) × Sp(1)) and the Cayley flag F_{4}/Spin(8) also admit isometric ℤ_{2} quotients. In summary all known simply connected, even dimensional manifolds with positive curvature admit isometric ℤ_{2} quotients if they do so topologically, since the remaining known examples are the compact, rank one, symmetric spaces (cf. [9]).

Acknowledgements

I would like to thank my advisor K.Grove for all his help, and in particular for suggesting that I investigate isometry groups of positively curved manifolds to look for free actions of finite groups.

References

[1] M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola. Norm. Sup. Pisa 15 (1961), 179-246.

[2] K. S. Brown, Cohomology of Groups, Springer-Verlag New York Inc., 1982.

[3] J.-H. Eschenburg, Inhomogeneous spaces of positive curvature, Diff.Geom.Appl. 2 (1992), 123-132.

[4] W. Klingenberg, Über Mannigfaltigkeiten mit positiver Krümmung, Comm. Math. Helv. 35 (1961), 47-54.

[5] I. Madsen, C. B. Thomas, C. T. C. Wall, The topological spherical space form problem– II, Topology 15 (1976), 375-382.

[6] P. Petersen, Comparison geometry problem list, Fields Inst. Monogr. 4, Amer. Math. Soc., Providence, RI, 1996.

[7] Proc. of the US-Japan Seminar in Differential Geometry, Kyoto, Japan (1965).

[8] X. Rong, The almost cyclicity of the fundamental groups of positively curved manifolds, Invent. Math. 126 (1996), no.1, 47-64.

[9] N. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math., 96 (1972), 277-295.

[10] B. Wilking, The normal homogeneous space (SU(3) × SO(3))/ U^{∗}(2) has positive sectional curvature, Proc. Amer. Math. Soc., to appear.

[11] J. A. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish Inc., 1984.

[12] S.-T. Yau, Seminar on Differential Geometry, Ann. Math. Studies, Princeton Univ. Press, Princeton, NJ, 1982.

Department of Mathematics, University of Maryland, College Park, MD 20742