Cohomology in algebra, geometry, physicsand statistics

Formally integrable complex structures on higher dimensional knot spaces

By the Brown-Gray’s classification, there are four classes of Riemannian manifolds M with parallel r-fold vector cross products: r = 1 and M a Kähler manifold, r = dim M − 1, r = 2 and M a G_2-manifold, r = 3 and M a Spin(7)-manifold. For the first three classes it has been proven by Brylinski, LeBrun, and Verbitsky, via ad hoc arguments for each of these classes, that the higher knot spaces for M carry a natural formally Kähler structure. More recently, Henrich provided a new proof for the r = dim M − 1 case. In a recent work with Hông Vân Lê (arXiv:1912.05175), we show how a variant of Henrich's construction can be used to provide a uniform proof for all four classes. In particular, this provides a proof for the previously unknown case of Spin(7)-manifolds.
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