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

TL;DR

This paper extends the classification of 1-connected 7-manifolds with torsion-free second homology, providing new proofs and applications involving Kreck-Stolz invariants and quadratic refinements.

Contribution

It generalizes a classification result for 1-connected 7-manifolds and applies it to 2-connected and specific cohomology ring cases with new proofs.

Findings

Classification of 1-connected 7-manifolds with torsion-free second homology.

New proof of a theorem by Crowley and Nordstroem for 2-connected 7-manifolds.

Application to simply connected 7-manifolds with specific cohomology rings.

Abstract

We generalize a result 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applications, one to 2-connected 7-manifolds (a new proof -- and slightly different formulation -- of an up to now unpublished Theorem by Crowley and Nordstroem and one to simply connected 7-manifolds with the cohomology ring of $S^2 \times S^5 \sharp S^3 \times S^4$. 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.