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

Abstract

We generalize a result [Kreck, Ann of Math. (2) 149 (1999) 707-754; Theorem 6] of the author about the classification of 1-connected 7-manifolds and demonstrate its use by two concrete applications, one to 2-connected 7-manifolds (a new proof -and slightly different formulationof an up to now unpublished Theorem by [Crowley and Nordström, Preprint, 2014, arXiv: 1406.2226) and one to simply connected 7-manifolds with the cohomology ring of S 2 × S 5 S 3 × S 4 . The answer is in terms of generalized Kreck-Stolz invariants, which in the case of 2-connected 7-manifolds is equivalent to a quadratic refinement of the linking form and a generalized Eells-Kuiper invariant.

The classification of 2‐connected 7‐manifolds

“…In this paper, we present a comprehensive smooth classification of closed 2‐connected 7‐manifolds by defining a generalisation of the Eells–Kuiper invariant for all spin 7‐manifolds. The uniqueness part of this classification has also been proven by Kreck [, Theorem 1]. Our main classification results are stated in Sections 1.2 and 1.3, and Section 1.4 describes the definition of the generalised Eells–Kuiper invariant.…”

“…Until recently the relevant results from modified surgery came from working over the normal 2‐type and rested on the general classification theorem [, Theorem 6], which in the 2‐connected case makes the very restrictive hypothesis that is generated by . In , Kreck defines an enhanced normal 2‐type which applies to all 2‐connected and which he uses to give an alternative proof of the uniqueness part of Theorem . The enhanced normal 2‐type encodes what Kreck calls a ‐structure which, in the notation of this paper, is a pair , where .…”

“…We would like to thank Jim Davis, Sebastian Goette, Matthias Kreck and Claude LeBrun for helpful comments and/or discussions. We would also like to thank Matthias Kreck for pointing out a mistake in the formulation of Theorem 1.4 in the first version of this paper and for sharing drafts of his paper [32]; in particular the explicit form of the transformation rule in (1) is based on [32, (2)]. In addition, we thank the referee for their careful comments, which have improved the paper.…”

On the classification of certain 1-connected 7-manifolds and related problems

“…To see that nondiffeomorphic Σ 7 leads to nondiffeomorphic M ′ k,0 #Σ 7 , we need a Kreck-Stolz type invariant s(M ). It is essentially the invariant s 1 defined in [13,Section 4]. We will give a brief review of it.…”

“…It has been proved in [8] that the total spaces of these bundles admit Riemannian metrics with nonnegative sectional curvature, hence are important to geometers. Relevant work concerning 7-manifolds of this homology type is [13], where the manifolds are required to have integral cohomology ring isomorphic to that of S 2 × S 5 #S 3 × S 4 .…”