Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:36:33Z http://mathoverflow.net/feeds/question/97377 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97377/codimension-two-embeddings-in-goodwillie-weiss-manifold-calculus-and-the-difficu Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups Hiro Lee Tanaka 2012-05-19T06:53:48Z 2012-06-20T16:08:25Z <p>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 $\leq \dim N-3$. </p> <p>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.</p> <p>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?</p>