Spaces of knotted circles and exotic smooth structures

Gregory Arone,Markus Szymik

doi:10.4153/s0008414x2000067x

Gregory Arone, Markus Szymik

Open Access

https://doi.org/10.4153/s0008414x2000067x

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Abstract

AbstractSuppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$ -type. In the four-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 $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ . We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

Highlights

Suppose that N and N are closed smooth manifolds of dimension n that are homeomorphic
Oleg Viro asked: is the algebraic topology of the space of smooth -knots in a -manifold sensitive to the smooth structure on the ambient manifold [ ]? More generally: can the homotopy type of the embedding space Emb(S, N) of knotted circles in a manifold N detect exotic smooth structures on N? One of our main results answers these negatively in a range: eorem A Let N be a smooth manifold of dimension n. e homotopy ( n − )-type of the space Emb(S, N) of smooth embeddings of the circle into N does not depend on the smooth structure
In dimension n =, which is the context of Viro’s original question, our result says that the spaces of knotted circles in two homeomorphic -manifolds have sets of components that are in bijection and that the corresponding components have isomorphic fundamental groups

Summary

Linear approximation: immersions

We point out that while, in general, Imm(M, N) is sensitive to the smooth structure on N, the space Imm(S , N) is not. is is true for target manifolds N of all dimensions. If smooth manifolds M and N are homeomorphic, the total spaces of the spherical tangent bundles S(M) and S(N) are homotopy equivalent. E homotopy type of the space Imm(S , N) of immersion of the circle into a smooth manifold N does not depend on the smooth structure of N. The Goodwillie–Klein result implies the much more elementary fact that the inclusion Emb(M, N) → Imm(M, N) is (n − m − )-connected. It follows for M = S that the inclusion Emb(S , N) → Imm(S , N) is (n − )-connected. Using well-known facts about the homotopy groups of πi Λ(N), we obtain the following proposition. If N is connected, the fundamental group of the space of immersions is isomorphic to π N ≅ H (N; Z)

Quadratic approximations

Embeddings of the circle

Applications to the fourth dimension