The classification of 2‐connected 7‐manifolds

Abstract

We present a comprehensive classification of closed smooth 2‐connected manifolds of dimension 7. This builds on the almost‐smooth classification from the first author's thesis. The main new ingredient is a generalisation of the Eells–Kuiper invariant that is defined for any closed spin 7‐manifold M, regardless of whether the spin characteristic class pM∈H4false(Mfalse) is torsion. We also determine the inertia group of 2‐connected M — equivalently the number of oriented smooth structures on the underlying topological manifold — in terms of pM and the torsion linking form.

Positive Ricci curvature on highly connected manifolds

“…In this case, the theorem is proven by Wall [Wall5,Theorem p. 156]. In the case where k ≡ 0 mod 2 this is proved in [C1,Lemma 3.12]. For the remaining case, k ≡ 1 mod 4, an extended quadratic form (H, λ, α) is an even form (H, λ) and a homomorphism α : H → Z 2 .…”

“…A few words are needed concerning Definition 5.7. Firstly, the sign of q M in Definition 5.7, differs from that given in [C1,Definition 2.10]. This is because [C1, Definition 2.22] gave the wrong sign for the linking of form b M .…”

On the classification of 1-connected 7-manifolds with torsion free second homology

“…We generalize a result [, Theorem 6] about the classification of 1‐connected 7‐manifolds and demonstrate its use by two concrete applications. The first application of our main theorem (Theorem ) is a new proof (and slightly different formulation) of an up to now unpublished Theorem by Crowley and Nordström about the classification of 2‐connected 7‐manifolds. To formulate the theorem in a convenient way we define the concept of a d‐structure .…”

“…The divisibility of in , denoted by (if , is torsion, we set ). For a spin 7‐manifold it was proven in [, Lemma 6.5], that the spin Pontrjagin class reduces mod 2 to the fourth Stiefel–Whitney class, which for spin‐manifolds is the fourth Wu‐class and by definition is 0 for dimensional reasons (see [, Lemma 2.2 (i)]). Thus is even.…”

“…We will later give formulas for the change of the invariant if we change the ‐structure (Proposition ). In , it was shown that the invariants used there can be realized.…”