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In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $Emb(M,N)$ \operatorname{Emb}(M,N)$is analytic in$M$if$dim \dim M \leq \dim N - 3$. Another example is given by the usual multiple disjunction lemma, which gives estimates on connectivity so long as we study the disjunction of manifolds$L_i$which have dimension$\leq \dim N-3$. At the same time, when I think of codimension 2 embeddings, I think of introducing$\pi_1$complications. (For instance, think of a 3-manifold, and removing a link. More simply: Remove a point from$\mathbb{R}^2$.) And as a general philosophy of topology, spaces with$\pi_1 \neq 0\$ are more difficult to study.

This is a somewhat vague question: Are these two complications related in an obvious or philosphical way, deeper than what I've said here? That is, is there a specific sense in which the codimension-two complications of manifold calculus are a manifestation of a general viewpoint, that non-simply-connected spaces are complicated?

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# Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and Thethe Difficulty of Fundamental Groups

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