Which Spin 7-Manifolds with S2×S5 Cohomology Admit Free Circle Actions?

Exactly 462 of 672 types admit a free circle action

The paper's first main result answers the existence question by counting. Of the 672 oriented diffeomorphism types of spin cohomology S2×S5 manifolds, precisely 462 admit a free circle action. The remaining 210 do not.

That count is not evenly spread across coarser classifications. The 672 oriented diffeomorphism types fall into 24 oriented homeomorphism types, each containing 28 oriented diffeomorphism types, and into 18 oriented homotopy types, of which 6 contain 2 oriented homeomorphism types and 12 contain a single one. The 462 admitting a free circle action cover 18 of the 24 oriented homeomorphism types and 12 of the 18 oriented homotopy types.

The paper also records the unoriented count: since the existence of a free circle action does not depend on orientation, precisely 235 of the 340 unoriented diffeomorphism types admit one. The oriented category is the one analysed in detail because the s-invariants are more convenient there.

Spin cohomology S2×S5 is a cohomology condition, not a homeomorphism

A spin cohomology S2×S5 is defined as a smooth, closed, simply-connected spin 7-manifold whose cohomology ring is isomorphic to that of S2×S5. This is a strictly weaker condition than being homeomorphic to S2×S5, and the distinction drives the whole count.

The paper contrasts its setting with an earlier result: all 28 oriented diffeomorphism types of closed 7-manifolds homeomorphic to S2×S5 admit a free circle action. The 672 spin cohomology S2×S5 types are a larger class, and only 462 of them admit such an action.

The 672 types split into two subfamilies of 336 each: those with π4(M) ≅ Z/2, which includes M = S2×S5, and those with π4(M) = 0. This split matters later, because the π4(M) = 0 subfamily behaves uniformly well.

A free circle action is the same data as a principal circle bundle

The paper works with the standard equivalence: a manifold admits a free circle action exactly when it is the total space of a principal circle bundle over a manifold of one dimension lower. Two free circle actions are equivalent exactly when there is a diffeomorphism of the base spaces pulling one Euler class to the other.

This reformulation turns the existence question into a question about which 6-manifolds can appear as orbit spaces and which Euler classes they can carry. The paper first determines which closed, simply-connected 6-manifolds can be the orbit space of a free circle action on a spin cohomology S2×S5, then computes the s-invariants of the total spaces of principal circle bundles over those 6-manifolds.

The classification of the 462 manifolds is stated in terms of their Kreck–Stolz s-invariants. Because the resulting formulae are highly complicated, the paper uses a computer to verify explicitly that each manifold claimed in the classification can be realised this way; the results are tabulated in the appendix.

Existence always comes with infinitely many non-equivalent actions

The second part of the main theorem is a multiplicity statement rather than an existence statement. Whenever a spin cohomology S2×S5 admits a free circle action, it admits infinitely many pairwise non-equivalent free circle actions.

The mechanism is visible in the proof. For a manifold satisfying the relevant hypotheses, there are infinitely many principal circle bundles over a family of closed, simply-connected 6-manifolds that are pairwise non-diffeomorphic. Because the base spaces are pairwise non-diffeomorphic, the induced circle actions on the total spaces are pairwise non-equivalent.

The paper notes that this multiplicity statement appears to be new even in the simplest case of S2×S5 itself.

441 types carry an invariant positive Ricci metric

The geometric result is narrower than the topological one. There exist 441 oriented diffeomorphism types of spin cohomology S2×S5 manifolds that admit infinitely many pairwise non-equivalent free circle actions and, for each of these actions, an invariant Riemannian metric of positive Ricci curvature. These include all 28 oriented diffeomorphism types homeomorphic to S2×S5.

The gap between 462 and 441 is 21 oriented diffeomorphism types. These admit a free circle action by the first theorem but are not covered by the metric construction, because they only have free circle actions with non-spin orbit space. Whether they admit an invariant Riemannian metric of positive Ricci curvature is left open.

The construction uses a result of Gilkey–Park–Tuschmann: if a closed manifold has finite fundamental group and its quotient admits a Riemannian metric of positive Ricci curvature, then the manifold admits an invariant Riemannian metric of positive Ricci curvature. The base spaces in the construction all admit such metrics, so the total spaces inherit them.

An important qualification: the paper does not claim that for these manifolds any free circle action admits an invariant Riemannian metric of positive Ricci curvature. The claim is that for each of the constructed actions such a metric exists.

When π4(M) = 0, every case works

One subfamily behaves without exception. If M is a spin cohomology S2×S5 with π4(M) = 0, then M admits infinitely many pairwise non-equivalent free circle actions and, for each of these actions, an invariant Riemannian metric of positive Ricci curvature.

This is a direct consequence of the classification results combined with the π4 split of the 672 types into two subfamilies of 336. The π4(M) = 0 half is fully covered by the metric construction; the difficulty is concentrated in the other half.

The 420 oriented diffeomorphism types that admit a free circle action with spin orbit space are all included in the metric result. The remaining 21 types from the 462 are exactly those whose only free circle actions have non-spin orbit space.

Connected sums of S2×S5, S3×S4 and homotopy spheres

The paper extends the construction to connected sums. Let Σ be a homotopy 7-sphere and let M = #ℓ(S2×S5) # m(S3×S4) # Σ. If ℓ ≥ 2, or ℓ = 1 and m is even, then M admits infinitely many pairwise non-equivalent free circle actions and, for each of these actions, an invariant Riemannian metric of positive Ricci curvature.

The existence of a free circle action on these manifolds was already known. What is new here is the invariant positive Ricci metric: when Σ is not the standard sphere, the existence of such a metric, even under a single free circle action, was not previously in the literature.

The paper also shows why the base-space search is delicate. If M is a spin cohomology S2×S5 and its orbit space N splits non-trivially as a connected sum N = N1 # N2, then both N1 and N2 are homotopy CP3s and M is homeomorphic to S2×S5. So the interesting orbit spaces do not decompose.

The 21-type gap is the paper's sharpest open edge

The most decision-relevant split in this paper is not the headline 462 versus 672. It is the 441 versus 462 gap. The topological existence question is settled by the classification in terms of Kreck–Stolz s-invariants, and the multiplicity statement holds uniformly whenever an action exists. The geometric question is where the boundary lies.

For a reader comparing manifolds in this class, the operative distinction is whether the manifold admits a free circle action with spin orbit space. If it does, it falls inside the 441 and the invariant positive Ricci metric is available. If it does not, it is one of the 21 types whose only free circle actions have non-spin orbit space, and the metric question remains open.

The π4(M) = 0 subfamily is the clean case: all 336 of its oriented diffeomorphism types are covered. The π4(M) ≅ Z/2 subfamily, which contains S2×S5 itself, is where the 21-type gap sits. That is the specific place where the paper's results stop and the next question begins.

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