I will present a program to classify closed, simply-connected, spin 7-manifolds M. The classification is organised by the groups H_2(M) = A and H_3(M), Due to the work of Kreck, Kreck-Stolz, Hepworth and others, much is known when A is torsion free.
It is helpful to first classify ``polarised manifolds", where an isomorphism H_2(M) \to A is added to the data. Setting H_3(M) = 0, we obtain a set of “sphere-like” manifolds, which we call ``\Theta-manifolds”, and taking the subset of polarised \Theta-manifolds which spin bound over the Eilenberg-MacLane space K(A, 2), we obtain the group bspin_8(A).
Our program has 4 mains steps: compute the group bspin_8(A), prove that it acts on sets isomorphism classes of polarised manifolds, identify invariants which classify the orbit of this action, and finally show how to compute the stabiliser of each orbit.
This is part of a joint project with Johannes Nordström, and includes joint work with Csaba Nagy.