Topological methods in enumerative geometry of G2 manifolds

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We propose to study a weak, homotopy-theoretical notion of associative submanifolds in G2 manifolds. This will allow us to apply powerful tools from algebraic topology, by-passing the substantial analytic difficulties one typically encounters in calibrated geometry. We will also study a comparison map relating a certain space of associative submanifolds for all G2-metrics in a given homotopy class of G2-structures to the space of homotopy-theoretic associatives. From this, we expect to gain new insights into existence and counting questions for associatives.