A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture

Thomas Schick^{∗} Fachbereich Mathematik — Universität Münster Einsteinstr. 62 48149 Münster

June 24, 1997

Abstract Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with π_{1}(M) ≅ ℤ^{4} × ℤ/3, so that the index invariant in the KO-theory of the reduced C^{∗}-algebra of π_{1}(M) is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of positive scalar curvature. The existence of such a metric is predicted by the (unstable) Gromov-Lawson-Rosenberg conjecture.

1 Obstructions to positive scalar curvature

1.1. Definition. A manifold M which admits a metric of positive scalar curvature is called a pscm-manifold.

We start with a discussion of the index obstruction for spin manifolds to be pscm, constructed by Lichnerowicz [7], Hitchin [4] and in the following refined version due to Rosenberg [10].

1.2. Theorem. Let M^{m} be a closed spin-manifold, π := π_{1}(M). One can construct a homomorphism, called index, from the singular spin bordism Ω_{∗}^{spin}(Bπ) to the (real) KO-theory of the reduced real C^{∗}-algebra of π:

ind : Ω_{∗}^{spin}(Bπ) → KO_{∗}(C_{red}^{∗}π)

Let u : M → Bπ be the classifying map for the universal covering of M. If M is pscm, then

ind([u : M → Bπ]) = 0 ∈ KO_{m}(C_{red}^{∗})

∗e-mail: thomas.schick@math.uni-muenster.de www: http://www.uni-muenster.de/math/u/lueck/ Fax: ++49 -251/83 38370

Gromov/Lawson [3] and Rosenberg [9] conjectured that the vanishing of the index should also be sufficient for existence of a pscm on M if m ≥ 5. This was proven by Stefan Stolz [15] for π = 1, and subsequently by him and other authors also for a few other groups [9, 6, 1, 11].

In dimension ≥ 5 there is only one additional obstruction for pscm, the minimal surface method of Schoen and Yau, which we will recall now. (In dimension 4, the Seiberg-Witten theory yields additional obstructions). The first theorem is the differential geometrical backbone for the application of minimal surfaces to the pscm problem:

1.3. Theorem. Let (M^{m}, g) be a manifold with positive scalar curvature, dim M = m ≥ 3. If V is a smooth m − 1-dimensional submanifold of M with trivial normal bundle, and if V is a local minimum of the volume functional, then V admits a metric of positive scalar curvature.

Proof. Schoen/Yau: [12, 5.1] for m = 3, [13, proof of Theorem1] for m > 3.

The next statement from geometric measure theory implies applicability of the previous theorem if dim(M) ≤ 7.

1.4. Theorem. [8, chapter 8] and references therein, in particular [2, 5.4.18] Suppose M^{m} is a smooth orientable closed manifold, dim M = m ≤ 7. Suppose 0 ≠ x ∈ H_{m−1}(M, ℤ). Then a smooth orientable closed m−1-dimensional submanifold V of M exists which represents x and which has minimal volume under all currents which represent x. In particular, V is a local minimum of the volume functional with orientable (hence trivial) normal bundle.

This implies the following statement about homology and cohomology which was observed by Stephan Stolz. Let X be any space.

1.5. Definition. For m ≥ 2 we define

H_{m}^{+}(X, ℤ) := {f_{∗}[M] ∈ H_{m}(X, ℤ); f : M^{m} → X and M is pscm}

1.6. Corollary. Let X be any space, α ∈ H^{1}(X, ℤ). Cap-product with α induces a map

α∩ : H_{m}(X, ℤ) → H_{m−1}(X, ℤ).

If 3 ≤ m ≤ 7, then α∩ maps H_{m}^{+}(X, ℤ) to H_{m−1}^{+}(X, ℤ).

Proof. If f : M^{m} → X represents x ∈ H_{m}^{+}(X, ℤ) and M is a pscm, then by theorems 1.3 and 1.4 the class f^{∗}α ∩ [M] is represented by N^{m−1} ↪^{j} M with a pscm N. In particular α ∩ x = f_{∗}(f^{∗}α ∩ [M]) is represented by f ∘ j : N → X, i.e. α ∩ x ∈ H_{m−1}^{+}(X, ℤ).

2 Counterexample to the Gromov-Lawson-Rosenberg conjecture

The aim of this paper is to present examples which show that the conjecture is not true for arbitrary fundamental groups.

To produce the counterexample, we use the only other known obstruction for pscm, namely the minimal surface method explained above.

The fundamental group will be π := ℤ^{4}×ℤ/3. We start with the computation of the KO-theory of the C^{∗}π-algebra. Note that the reduced C^{∗}-algebra of the product of two groups is the (minimal) tensor product of the individual C^{∗}-algebras [14, p. 14–15]. By [14, p. 14 and 1.5.4]

KO_{n}(C_{red}^{∗}(ℤ^{4} × ℤ/3)) ≅ ⨁_{i∈I} KO_{n−n_{i}}(C_{red}^{∗}(ℤ/3)); |I| < ∞

For finite groups, it is well known that their KO-theory is a direct sum of copies of the KO-theories of ℝ, ℂ and ℍ. In particular, it is a direct sum of copies of ℤ and ℤ/2. Therefore, the same is true for π:

2.1. Proposition. KO_{∗}(C_{red}^{∗}π) is a direct sum of copies of ℤ and ℤ/2. In particular, its torsion is only 2-torsion.

We will now construct a spin manifold M^{5} with π_{1}(M) = π, so that the class [u : M → Bπ] ∈ Ω_{5}^{spin} is 3-torsion. Then, automatically

ind(u : M → Bπ) = 0 ∈ KO(C_{red}^{∗}π)

2.2. Example. Let p : S^{1} → Bℤ/3 be a map so that π_{1}(p) is surjective and equip S^{1} with the spin structure induced from D^{2}. This is 3-torsion since Ω̃_{1}^{spin}(Bℤ/3) ≅ H_{1}(Bℤ/3, ℤ) ≅ ℤ/3 (use the Atiyah-Hiryebruch spectral sequence). Consider the singular manifold

f = id × p : S^{1} \overbrace{× · ×}^{4} S^{1} × S^{1} → S^{1} \overbrace{× · ×}^{4} S^{1} × Bℤ/3 = Bπ

This is then 3-torsion in Ω_{5}^{spin}(Bπ). Doing surgery we can construct a bordism F : W → Bπ in Ω_{5}^{spin}(Bπ) from f to some u : M → Bπ where u is an isomorphism on π_{1}.

Now, M is a manifold with trivial index, and we have to show that it is not pcsm. Assume that the converse is true.

We study the homology and cohomology of π first. By the Künneth theorem

\begin{aligned}H_{1}(Bπ, ℤ) & = x_{1}ℤ ⊕ ⋯ ⊕ x_{4}ℤ ⊕ yℤ/3 \\ H^{1}(Bπ, ℤ) & = a_{1}ℤ ⊕ ⋯ ⊕ a_{4}ℤ \\ 0 ≠ w = x_{1} × ⋯ × x_{4} × y & ∈ H_{5}(Bπ, ℤ) \\ 0 ≠ z = x_{4} × y = a_{1} ∩ (a_{2} ∩ (a_{3} ∩ w)) & ∈ H_{2}(Bπ, ℤ) \end{aligned}

We use the map

B_{∗} : Ω_{∗}^{spin}(X) → H_{∗}(X, ℤ) : [f : M → X] ↦ f_{∗}[M]

which is a edge homomorphism in the Atiyah-Hirzebruch spectral sequence. Of course, w = f_{∗}([T^{5}]) = u_{∗}[M] is the image of the considered singular manifold under this transformation.

If M would admit a pscm, then

w ∈ H_{5}^{+}(Bπ)

Iterated application of theorem 1.6 implies that

0 ≠ z ∈ H_{2}^{+}(Bπ)

But there is only one two dimensional oriented manifold with positive curvature, namely S^{2}. Since π_{2}(Bπ) = 0 any map g : S^{2} → Bπ is null homotopic. In particular g_{∗}[S^{2}] = 0 ∈ H_{2}(Bπ, ℤ), and therefore H_{2}^{+}(Bπ, ℤ) = 0.

This is the desired contradiction and M does not admit a metric with positive scalar curvature. The proof relies on the existence of torsion in π. Therefore, one may still conjecture that the

unstable Gromov-Lawson-Rosenberg conjecture is true for torsion free groups.

3 Acknowledgement

To work on the counterexample was inspired by talks of Stephan Stolz where he expressed his opinion that the original GLR-conjecture is false. The author wants to thank Stephan Stolz for useful and enlightening conversations on the subject. Stolz conjectures that a weaker form, the so called stable GLR-conjecture, is true (compare [16]) and shows [17] that this conjecture follows from the Baum-Connes conjecture. The group π fulfills the Baum-Connes conjecture. This follows e.g. from deep results of Julg/Kasparow [5, 8.1 and 8.5]. Therefore, M is a counterexample to the unstable Gromov-Lawson-Rosenberg conjecture but if B is a Bott manifold and n ist sufficiently large M × B^{n} admits a metric with positive scalar curvature.

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