Topological methods in enumerative geometry of G2 manifolds

我们建议研究G2流形中结合子流形的一个弱的同伦理论概念。这将允许我们应用代数拓扑学中的强大工具，绕过人们在校准几何中通常遇到的实质性分析困难。我们还将研究在给定的G2-结构同伦类中所有G2-度量的结合子流形空间与同伦论结合空间之间的比较映射。从这一点上，我们期望获得对存在和计算联想词问题的新见解。

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.