论文标题
打结空间和Atiyah二元模型
Models for knot spaces and Atiyah duality
论文作者
论文摘要
令$ \ mathrm {emb}（s^1，m）$为从圆的光滑嵌入到一个封闭的歧管$ m $ dimension $ \ geq 4 $的空间。我们使用poincaré-lefschetz二元性的光谱版本称为atiyah duality，研究了稳定类别中$ \ mathrm {emb}（s^1，m）$的cosimimplicial模型。实际上，我们处理了一个综合模型的概念，并证明了双重性的综合版本。作为一个应用程序，我们将新的频谱序列融合到$ h^*（\ Mathrm {emb}（s^1，m））$中，对于简单连接的$ m $和主要系数环。使用此情况，我们在低度中计算$ h^*（\ MATHRM {emb}（s^1，s^k \ times s^l））$，在某些情况下，$ k $，$ l $。我们还证明了包含$ \ MATHRM {emb}（s^1，m）\ to \ mathrm {imm {imm}（s^1，m）$ in Immersions诱发了与某些简单连接的$ 4 $ - manifolds的$π_1$相关的同构，与Arone and szymik相关的问题。我们还证明了$ \ mathrm {emb}（s^1，m）$的奇异科链复合物以及在可理解空间上的束束光谱的链谱群的同型。我们的关键要素是由于R. Cohen引起的双重性的结构化版本。
Let $\mathrm{Emb}(S^1,M)$ be the space of smooth embeddings from the circle to a closed manifold $M$ of dimension $\geq 4$. We study a cosimplicial model of $\mathrm{Emb}(S^1,M)$ in stable categories, using a spectral version of Poincaré-Lefschetz duality called Atiyah duality. We actually deal with a notion of a comodule instead of the cosimplicial model, and prove a comodule version of the duality. As an application, we introduce a new spectral sequence converging to $H^*(\mathrm{Emb}(S^1,M))$ for simply connected $M$ and for major coefficient rings. Using this, we compute $H^*(\mathrm{Emb}(S^1, S^k\times S^l))$ in low degrees with some conditions on $k$, $l$. We also prove the inclusion $\mathrm{Emb}(S^1,M)\to \mathrm{Imm}(S^1,M)$ to the immersions induces an isomorphism on $π_1$ for some simply connected $4$-manifolds, related to a question posed by Arone and Szymik. We also prove an equivalence of singular cochain complex of $\mathrm{Emb}(S^1,M)$ and a homotopy colimit of chain complexes of a Thom spectrum of a bundle over a comprehensible space. Our key ingredient is a structured version of the duality due to R. Cohen.
