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, $ \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.

假设 $N_1$ 和 $N_2$ 是同胚的维数n的封闭光滑流形。我们证明了光滑结的空间 $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ 和 $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1 , N_2),$ 具有相同的同伦 $(2n-7)$ 型。在四维情况下，这意味着同胚 $4$ -流形中的光滑结空间具有双射分量的集合 $\ pi_0$ ，并且对应的路径分量具有相同的基本群 $\pi_1$ . 关于 $\pi _0$ 的结果 是众所周知的和基本的，但关于 $\pi _1$ 的结果似乎是新的。结果对 Oleg Viro 的问题给出了否定的部分答案。我们的证明使用 Goodwillie-Weiss 嵌入塔。我们给出了Goodwillie-Weiss塔的二次级新模型，证明了N中节点空间二次逼近的同伦类型不依赖于N上的光滑结构。 我们的结果还给出了$\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ 的下限 。我们使用我们的模型表明，对于每个基点选择，每个同伦群 $\pi_1 $ 和 $\pi_2, $ $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ 包含一个无限生成的自由阿贝尔群。