Some Results on Operads and Configuration Spaces

The little <em>n</em>-disk operad <em>E</em>_n is constructed geometrically as the collection of configuration spaces of n-disks. This is not an isolated example of an interaction between operads and geometry: geometric objects often are either acted upon by operads, form operads, or have associated moduli spaces which form operads. One major area of study in geometric topology is the configuration spaces of nonoverlapping points in M, and more generally, embedding spaces of one manifold into another. In this thesis, we study these spaces from the perspective of operads in spectra. The category of spectra is the "linearization'' of the category of spaces, and as such it has a rich algebraic structure which parallels that of the category of chain complexes. As a result, many classical techniques from from the homological algebra of chain complexes can be used. One such technique is Koszul duality for operads, modules, and algebras.

We initially study the stabilization of configuration space as an object in spectra, computing its Spanier-Whitehead dual and deducing the homotopy invariance of stabilized configuration space. For a framed manifold <em>M</em> of dimension <em>n</em>, the collection of all configuration spaces Conf(<em>M,i</em>) has a right module action of the E_n operad. We study the E_n operad action on the collection of configuration spaces and deduce results about Browder brackets in the homology of configuration spaces. We also deduce homotopy invariance results concerning the associated graded of filtrations of stabilized embedding spaces. Ultimately, we prove that the stabilized configuration spaces of a framed manifold are a fixed point of Koszul duality for right modules. In other words, the collection of stabilized configuration spaces of a framed manifold is Koszul self dual. This statement generalizes Atiyah duality to include the action of the E_n operad and has applications to spaces of embeddings and configuration spaces with labels in an E_n-algebra.

小n-圆盘操作子（operad）E_n 可通过几何方式构造为n圆盘的配置空间的集合。这并非操作子与几何学之间相互作用的孤立案例：几何对象通常要么被操作子作用，要么自身构成操作子，或是其关联的模空间构成操作子。几何拓扑学的核心研究方向之一，便是流形M内不重叠点的配置空间，更一般地则是单一流形到另一流形的嵌入空间。本文从谱范畴内的操作子视角出发，对上述空间展开研究。谱范畴是拓扑空间范畴的“线性化”，其拥有与链复形范畴高度相似的丰富代数结构。正因如此，诸多源自链复形同调代数的经典技术均可得到应用，其中之一便是针对操作子、模与代数的科斯祖尔对偶（Koszul duality）。

我们首先将配置空间视为谱范畴中的对象，研究其稳定化过程，计算其斯潘尼尔-怀特海德对偶（Spanier-Whitehead dual），并推导出稳定化配置空间的同伦不变性。对于n维带标架流形M，全体配置空间Conf(M,i)的集合带有E_n操作子的右模作用。我们研究E_n操作子在配置空间集合上的作用，并推导出配置空间同调中的布劳德括号（Browder brackets）相关结论。此外，我们还推导出关于稳定化嵌入空间滤过的相伴分次空间的同伦不变性结果。最终，我们证明了带标架流形的稳定化配置空间是右模上科斯祖尔对偶的不动点。换言之，带标架流形的稳定化配置空间集合是科斯祖尔自对偶的。这一结论将阿蒂亚对偶（Atiyah duality）推广至包含E_n操作子的作用，可应用于嵌入空间以及带有E_n代数标签的配置空间。