A Counterexample to a Conjecture about Positive Scalar Curvature

Our counterexample is based on the counterexample to the Gromov-Lawson-Rosenberg conjecture given in [5]. There, a 5-dimensional connected closed spin manifold M with fundamental group Γ = Z 4 ⊕ Z/3 is constructed, whose Rosenberg index vanishes but which nevertheless does not admit a metric of positive scalar curvature. By taking the connected sum of this manifold M with a simplyconnected non-spin manifold N , we obtain a totally non-spin manifold X which has the same fundamental group as M . One has BΓ = T 4 × BZ/3 and analogously BΓ = T 4 by [1, (1) and (4), p. 1624]. Especially, H n (BΓ) = 0 for n ≥ 5, so that the condition on f * [X] from Conjecture 1.1 is satisfied in the case at hand. The argument in [5] relies on the following observation by Stolz and we will also make significant use of this result. Lemma 1.2. Let X be a topological space and set for n ∈ N ≥2 H + n (X) := {f * [M ] ∈ H n (X) ; f : M n → X and M admits a metric with scal > 0} Then for any class u ∈ H 1 (X) the map [3,Theorem 4.4] for n = 8.

Our result reads now as follows.

Proposition 1.3. Let M be the manifold constructed in [5] (we recall its construction in Section 2) and N a simply connected manifold of dimension 5 which admits no spin structure. Then the manifold X := M #N has non-spin universal covering and admits no metric with positive scalar curvature.

This result is part of the first named author’s forthcoming thesis [4].

Proof of Proposition 1.3. First of all, if X is constructed as above, we have already noted that it has non-spin universal covering. To obtain an explicit simplyconnected non-spin 5-manifold N , one can start with CP 2 ×S 1 , which is non-spin as CP 2 is, and then do surgery on the embedded S 1 to obtain the simply-connected N . Because this surgery does not touch the embedded CP 1 with its non-spin normal bundle, the resulting N remains a non-spin manifold.

In order to see that X admits no metric of positive scalar curvature, we use the same argument as in [5]. To begin with, we choose the model BΓ = T 4 × BZ/3. Recall,

n even. Together with the Künneth formula this gives

Here we have written T 4 = X 1 × • • • × X 4 as product of four copies of S 1 , and X 5 for BZ/3.

Fix a basepoint x = (x 1 , . . . , x 5 ) ∈ BΓ and let p : T → BZ/3 be a map which induces an epimorphism on π 1 as in [5], as well as f j : X j → BΓ the map which includes X j identically and basepoint-preserving. We denote by [ * ] ∈ H 0 (BΓ) the canonical generator. Next, choose for each 1 ≤ j ≤ 4 generators g j ∈ H 1 (X j ) and elements g * j ∈ H 1 (X j ) with g * j , g j = 1, and let

is the standard generator for H 1 (S 1 ). Introduce the elements v j := (f j ) * (g j ) ∈ H 1 (BΓ) for j = 1, . . . , 5 as well as a 1 , . . . , a 4 ∈ H 1 (BΓ) with

Finally, set

. By the Künneth formula, w = 0 and z = 0. Furthermore,

For example one has

] as fundamental class for T 5 . Then f * [T 5 ] = w. As in [5] one can construct a bordism in Ω spin 5 (BΓ) from f to a map g : M → BΓ which induces an isomorphism of fundamental groups. This defines the manifold M . Now let N be any simply-connected closed non-spin manifold of dimension 5 and set X := M #N .

Finally, assume that X admits a metric of positive scalar curvature. Then consider the map h : M ⊔ N → BΓ on the disjoint union of M and N , which equals g on M and sends N to a point. One has h * [M ⊔ N ] = g * [M ] = w and since M ⊔ N is bordant to M #N , it follows that w ∈ H + 5 (X). But then it follows from ( * ) as well as Lemma 1.2 that w is mapped to z under the following composition

] for some k : S 2 → BΓ since S 2 is the only orientable surface which admits a metric of positive scalar curvature. On the other hand, π 2 (BΓ) = 0 so that k is null homotopic. This implies z = 0, which is a contradiction.

Remark 2.1. The method described in this note produces a counterexample to Conjecture 1.1 with fundamental group Γ whenever Γ satisfies the following homological conditions:

• for 5 ≤ m ≤ 8 there is a homology class [M ] ∈ H m (BΓ; Z) represented by an m-dimensional closed oriented manifold M (with surgeries one can then arrange that π

• under the map H m (BΓ) → H m (BΓ) the class [M ] is send to 0. Note that this condition is similar, indeed much easier than the general homological condition for counterexamples to the Gromov-Lawson-Rosenberg condition derived in [2]. Unfortunately, its structure requires the group Γ to contain non-trivial torsion, to allow for a kernel of the map H * (BΓ) → H * (BΓ) (in contrast to [2]).

The assumption on H 1 (BΓ; Z) we have to make is very strong, it has to have rank at least m -2. In particu

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