Spaces of knotted circles and exotic smooth structures

Abstract

Suppose that N 1 and N 2 are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots Emb(S 1 , N 1) and Emb(S 1 , N 2) have the same homotopy (2n − 7)-type. In the 4-dimensional case this means that the spaces of smooth knots in homeomorphic 4-manifolds have sets π 0 of components that are in bijection, and the corresponding path components have the same fundamental groups π 1. The result about π 0 is well-known and elementary, but the result about π 1 appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie-Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie-Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on π 2 Emb(S 1 , N). We use our model to show that for every choice of basepoint, each of the homotopy groups π 1 and π 2 of Emb(S 1 , S 1 × S 3) contains an infinitely generated free abelian group.

Embedding calculus and smooth structures

“…We investigate the scope of a certain tool used to study the space Emb s .N; M / of smooth embeddings from an n-manifold N into an m-manifold M . This investigation has consequences for spaces of embeddings themselves, as shown by the following result on knots and links, which answers a question of Viro [42,Section 6] in the negative and improves on a result of Arone and Szymik [3].…”

“…Remark 4.4 Our proof of Theorem A implies that, under the same hypotheses, the finite stages T r Emb s F k S 1 ; N and T r Emb s F k S 1 ; M are also weakly equivalent. A related result appears in [3,Theorem A], where a study of the second stage of the Taylor tower is leveraged to show that, if N is n-dimensional, the .2n 7/-skeleton of Emb s .S 1 ; N / does not depend on the smooth structure of N .…”

“…We now "thicken" all the submanifolds involved to codimension 0. Fixing a small > 0, we replace D 0 by D 3 We now show that embedding calculus captures this homotopy type; specifically, the left-hand vertical map is a weak equivalence in the commuting diagram Emb s @ 0 [@ 1 .C; D 3 n V D 3 / Emb s @ 1 .D 1 C; ; D 3 / Emb s .D 3 ; D 3 / T 1 Emb s @ 0 [@ 1 .C; D 3 n V D 3 / T 1 Emb s @ 1 .D 1 C; ; D 3 / T 1 Emb s .D 3 ; D 3 / giving an example of convergence in codimension 2. Since D 3 has handle dimension 0, isotopy extension for embedding calculus -or, rather, the extension to neat embeddings of manifolds with corners -implies that the bottom row is also a fibration sequence, so it suffices to show that the middle and right-hand vertical maps are weak equivalences, both of which follow from the Yoneda lemma.…”

On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

“…the second stage of the embedding calculus tower, see Remark 2.2), and a result of Longoni and Salvatore [LS05] shows that those spaces can distinguish nondiffeomorphic (3-)manifolds. Let us point out, however, that by [AS20] these extensions agree for homeomorphic smooth manifolds.…”

Models for knot spaces and Atiyah duality

“…Let Emb.S 1 ; M / be the space of smooth embeddings from the circle S 1 to a manifold M (without any basepoint condition) endowed with the C 1 -topology. The space Emb.S 1 ; M / is studied by Arone and Szymik [1] and Budney [8], and study of embedding spaces including the knot space is a motivation of Campos and Willwacher [10] and Idrissi [22] In the rest of the paper, M denotes a connected closed smooth manifold of dimension d . Our knot space Emb.S 1 ; M / is slightly different from the one considered by Sinha, but we can construct a cosimplicial model similar to Sinha's, which is called Sinha's cosimplicial model and denoted by C .M /.…”