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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?

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    $\begingroup$ The typical reader of this question will assume that "manifold calculus" is something like "calculus on manifolds", whereas actually you are referring to one of the several versions of this pet idea of mine called the "calculus of functors". But the really funny thing is that even I somehow failed to realize what the question was about from just looking at the title! I will try to come up with a sensible answer, but not this minute. $\endgroup$
    – Tom Goodwillie
    Commented May 19, 2012 at 17:16
  • $\begingroup$ That's really amusing -- I didn't realize! The first sentence of the question also could be a question about usual calculus (on manifolds). I've changed the title to be more explicit. Thanks, Tom. $\endgroup$
    – Hiro Lee Tanaka
    Commented May 20, 2012 at 0:15
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    $\begingroup$ I just wish to remark that the current proof of the multiple disjunction statements for smooth embeddings involve several steps: 1. reduction to statements about block embeddings (using Tom's thesis). 2. reduction of the latter to statements about Poincare embeddings (using surgery theory). 3. Proof of the statements about Poincare embeddings from (using homotopy theory). Each of these steps requires the codimension $\ge 3$ hypothesis somewhere along the way. $\endgroup$
    – John Klein
    Commented May 20, 2012 at 2:43
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    $\begingroup$ Yes, and 2 and 3 use the fact that the fundamental group is unchanged by cutting out something of codim $\ge 3$, though 1 does not. $\endgroup$
    – Tom Goodwillie
    Commented May 20, 2012 at 5:09

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