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# 数学代写|代数拓扑代考Algebraic Topology代考|Math215 Introduction and Overview

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## 数学代写|代数拓扑代考Algebraic Topology代考|Introduction and Overview

This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics. It aims to present both the theory of higher Hochschild (co)homology and its application to higher string topology. It contains detailed proofs of results stated in the note [Gi3] as well as some new results building on our previous work [GTZ3, Gi4] notably. One of the main new result is an application of the techniques of Higher Hochschild (co)homology to study higher string topology ${ }^1$ and prove that, in addition to its already rich algebraic package, the latter inherits an additional Hodge filtration (compatible with the rest of the structure). We also prove that the $E_n$-centralizer of maps of commutative (dg-)algebras are equipped with a Hodge decomposition and a compatible structure of framed $E_n$-algebras ${ }^2$ and study Hodge decompositions suspensions and products by spheres generalizing the ones of [P] and dual results of [TW], see below for more details on these results.

This various results are also a pretext to illustrate the techniques of higher order Hochschild Homology in the case of commutative differential graded algebras, both using its derived (in an $\infty$-categorical sense) interpretation and functoriality and emphasizing on and using its nice combinatorial structure and how to use it. The emphasis on this latter point is another benefit of this paper compared to most of the literature we know ${ }^3$ and a good way to get a feeling on the behavior and benefits of higher Hochschild (co)homology, in, we hope, a gentle way.

Higher Hochschild (co)homology was first emphasized by Pirashvili in [P] in order to understand the Hodge decomposition of Hochschild homology and how to generalize it. Higher Hochschild (co)homology is in fact a joint invariant of both topological spaces (or their homotopy combinatorial avatar : simplicial sets) and commutative differential graded algebras (CDGA for short). As the name suggests, it is a generalization for commutative (dg-)algebras of the standard Hochschild homology of dg-associative algebras. It is also a special case [GTZ2, AF] of factorization homology ${ }^4[\mathrm{BD}, \mathrm{Lu} 3, \mathrm{AF}]$ which get extra-functoriality and is one of the easiest one to compute and manipulate. ${ }^5$

## 数学代写|代数拓扑代考Algebraic Topology代考|Notations, Conventions and a Few Standard Facts

We fix a ground field $k$ of characteristic 0 . We will also use the following notations and conventions

If $\left(C, d_C\right)$ is a cochain complex, $C[i]$ is the cochain complex such that $C[i]^n:=$ $C^{n+i}$ with differential $(-1)^i d_C$. We will mainly work with cochain complexes and adopt the convention that a chain complex is a cochain complex with opposite grading when we need to compare gradings.

An $\infty$-category will be a complete Segal space. Any model category gives rise to an $\infty$-category.

We write k-Mod ${ }^{d g}$ for the category of cochain complexes and k-Mod for its associated $\infty$-category. We will use the abbreviation dg for differential graded. We will use the words (co)homology for an object of these $\infty$-categories (in other words a complex thought up to quasi-isomorphism) and use the words (co)homology groups for the actual groups computed by taking the quotient of the (co)cycles by (co)boundaries (for instance see Definition 1.3.19).

sSet and Top: sSet is the (model) category of simplicial sets, that is functors from $\Delta^{o p} \rightarrow$ Set where $\Delta$ is the simplex category of finite sets $n_{+}:={0, \ldots, n}$ with order preserving maps. We also have the (model) category of topological spaces Top. These two categories are Quillen equivalent: $|-|:$ sSet $\underset{\leftrightarrows}{\leftrightarrows}$ Top : $\Delta_{\bullet}(-)$. Here $\Delta_{\bullet}:$ Top $\rightarrow$ sSet the singular set functor defined by $\Delta_n(X)=$ $\operatorname{Map}{\text {Top }}\left(\Delta^n, X\right)$, where $\Delta^n$ is the standard $n$-dimensional simplex, and $\left|Y{\bullet}\right|$ the geometric realization. Their associated $\infty$-categories, respectively denoted sSet and Top are thus equivalent.

These four $(\infty)$-categories are symmetric monoidal with respect to disjoint union.

## 数学代写|代数拓扑代考Algebraic Topology代考|Introduction and Overview

Pirashvili 在 [P] 中首先强调了更高的 Hochschild (共) 同源性，以了解 Hochschild 同源性的 Hodge 分解以及如何推广它。 高等 Hochschild (共) 同源性实际上是拓扑空间 (或其同伦组合化身：单纯集) 和交换微分梯度代数（简称 CDGA) 的联合不变: 量。顾名思义，它是 dg-结合代数的标准 Hochschild 同调的交换 (dg-) 代数的推广。也是分解同调的特例[GTZ2,AF] [ $[\mathrm{BD}, \mathrm{Lu} 3, \mathrm{AF}]$ 它具有额外的功能性，是最容易计算和操作的一种。

## 数学代写|代数拓扑代考Algebraic Topology代考|Notations, Conventions and a Few Standard Facts

sSet 和 Top: sSet 是单纯集的 (模型) 类别，即来自 $\Delta^{o p} \rightarrow$ 设置位置 $\Delta$ 是有限集的单纯形范帱 $n_{+}:=0, \ldots, n$ 带有保留地图的 顺序。我们还有拓扑空间 Top 的 (模型) 类别。这两个光别是 Quillen 等效的: $|-|$ : 集士最佳： $\Delta_{\bullet}(-)$. 这里 $\Delta_{\bullet}$ :最佳 $\rightarrow$ sSet 定义的奇异集函子 $\Delta_n(X)=\operatorname{Map} \operatorname{Top}\left(\Delta^n, X\right)$ ，在哪里 $\Delta^n$ 是标准 $n$-维单纯形，和 $|Y \bullet| 几$ 何实现。他们的关联 $\infty$-类别，分 别表示为 sSet 和 Top，因此是等价的。

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