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

2012-05-19T06:53:48Z

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \leq \dim N - 3$. 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?

Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups - MathOverflow

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