Spaces of knotted circles and exotic smooth structures

TL;DR

This paper investigates the homotopy types of spaces of smooth knots in homeomorphic manifolds, revealing invariance under smooth structure changes and providing new insights into their fundamental groups and higher homotopy groups.

Contribution

It introduces a new model for the quadratic stage of the Goodwillie-Weiss tower and shows the homotopy type of the quadratic approximation is independent of the smooth structure.

Findings

Homotopy (2n-7)-type of knot spaces is the same for homeomorphic manifolds.

The fundamental group of knot spaces in 4-manifolds is invariant and contains infinitely generated free abelian groups.

Provides a lower bound on the second homotopy group of knot spaces in certain manifolds.

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 $\pi_0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi_1$. The result about $\pi_0$ is well-known and elementary, but the result about $\pi_1$ 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 $\pi_2 Emb(S^1, N)$. We use our model to show that for every choice of basepoint, each of the homotopy groups $\pi_1$ and $\pi_2$ of $Emb(S^1, S^1\times S^3)$ contains an infinitely generated free abelian group.