Manifold Calculus and Convex Integration

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Johns Hopkins University
The spaces of embeddings of one manifold inside another, are some of the most fundamental geometric objects studied in algebraic topology, the classical problem of studying knots embedded in R3 being the most well studied example. However, embedding spaces are extremely difficult to study. Most algebro-geometric techniques either produce a series of simpler spaces approximating them or they stabilize the space of embeddings in a suitable sense and study the stabilized version using stable homotopy theory. Manifold calculus analyzes embedding spaces as a contravariant functor from some categories of smooth manifolds to topological spaces and con- structs polynomial approximations to them, analogous to the way polynomial approximations can be constructed to a smooth function using Taylor series. Furthermore, the fundamental theorem of manifold calculus says that these approximations converge to the original functor. In this thesis, we apply Gromov’s theory of convex integration to study embedding spaces of manifolds with tangential structures. We prove that the theory of convex integration is compatible with the theory of manifold calculus, in the sense that the polynomial approximations produced by manifold calculus converge to the structured embedding spaces whenever the theory of ii convex integration can be applied. This also allows us to construct examples of functors whose polynomial approximations do not converge to the original functor. When N is a symplectic manifold, we prove that the analytic approxi- mation to the Lagrangian embeddings functor EmbLag(−, N) is the totally real embeddings functor EmbTR(−, N). When M ⊆ Rn is a parallelizable manifold, we provide a geometric construction for the homotopy fiber of Emb(M, Rn) → Imm(M, Rn). This construction also provides an example of a functor which is itself empty when evaluated on most manifolds but whose analytic approximation is almost always non-empty.
homotopy theory, embedding spaces, manifold calculus, convex integration, symplectic geometry