Existence of nonspin orbit spaces for free circle actions on kS^2×S^5 # lS^3×S^4 # Σ

Determine, for integers k,l ≥ 0 and any homotopy 7-sphere Σ, whether there exists a free smooth circle action on the 7-manifold kS^2 × S^5 # lS^3 × S^4 # Σ whose orbit space N is nonspin (i.e., w2(N) ≠ 0).

Background

The paper classifies when the 7-manifolds kS^2×S^5 # lS^3×S^4 # Σ admit free circle actions, and constructs such actions with spin orbit spaces using suspension operations. In several cases, the authors explicitly construct free circle actions with spin orbit, but they do not determine in general whether nonspin orbit spaces occur.

For related special cases, the paper gives partial results: for M = S^2×S^5 # Σ_r, a free circle action with nonspin orbit exists if and only if r is even; and for even k, any spin cohomology kS^2×S^5 admits no free circle action with nonspin orbit. The general existence of nonspin orbit spaces for arbitrary k,l,Σ remains unresolved.