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 ${\pi }_0$ 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.

20 pages. Our results on the example Emb( S^1, S^1xS^3 ) are now valid for every choice of basepoint

References in corpus (3)

• A family of embedding spaces
• Multiple disjunction for spaces of smooth embeddings
• Knotted 3-balls in S^4

Cited by in corpus (6)

• Knotted 3-balls in S^4
• Embedding calculus and smooth structures
• Theta-graph and diffeomorphisms of some 4-manifolds
• On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant
• Models for knot spaces and Atiyah duality
• On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold