Etale homotopy and sums-of-squares formulas

非阿贝尔基变换定理与étale同伦论

Nonabelian basechange theorems & étale homotopy theory

本文证明étale上同调中多个基变换定理的非阿贝尔精细化，并借此研究étale同伦型，证明其满足arc-descent并得到若干Künneth公式。

Motivic and Real Etale Stable Homotopy Theory

Let X be a Noetherian scheme of finite dimension and denote by rho the (additive inverse of the) morphism in SH(X) from S to Gm corresponding to the unit -1. Here SH(X) denotes the motivic stable homotopy category. We show that the category obtained by inverting rho in SH(X) is canonically equivalent to the (simplicial) local stable homotopy category of the site X_ret, by which we mean the small real etale site of X, comprised of etale schemes over X with the real etale topology. One immediate application is that SH(RR)[rho^-1] is equivalent to the classical stable homotopy category. In partic…

不稳定motivic与实étale同伦理论

Unstable motivic and real-étale homotopy theory

本文证明对任意基概形 $S$，其实étale motivic不稳定同伦理论与不稳定半代数拓扑一致，并指出带基点连通motivic空间的实étale motivic局部化由与映射 $ρ: S^0 \to {\mathbb G}_m$ 望远镜的smashing给出。

A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of étale homotopy. Using this model structure we define a pro-space associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated to a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [HaSc] for the study of obstructions t…

The étale symmetric Künneth theorem

Let $k$ be an algebraically closed field, $l\neq\operatorname{char} k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the étale homotopy type of the $d$th symmetric power of $X$ is $\mathbb Z/l$-homologically equivalent to the $d$th strict symmetric power of the étale homotopy type of $X$. We deduce that the $\mathbb Z/l$-local étale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.

连通和上同伦纤维化的有理同伦型

The Rational Homotopy Type of Homotopy Fibrations Over Connected Sums

本文研究连通和上拉回纤维化全空间的有理同伦型，给出较Jeffrey–Selick更弱的有理上同调条件，使其在取闭路后等价于连通和。

Existence of rational points as a homotopy limit problem

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of etale topological types under the Galois action. As our main example we show that the surjectivity statement in Grothendieck's Section Conjecture would follow from the surjectivity of the map from fixed points to continuous homotopy fixed points on the level of connected components. Along the way we define a new model for the continuous etale homotopy fixed point space of a smooth variety over a field under the Galois action.

Comparison of Stable Homotopy Categories and a Generalized Suslin-Voevodsky Theorem

Let $k$ be an algebraically closed field of exponential characteristic $p$. Given any prime $\ell\neq p$, we construct a stable étale realization functor $$\underline{\text{Ét}}_{\ell}:\text{Spt}(k)\rightarrow \text{Pro}(\text{Spt})^{H\mathbb{Z}/\ell}$$ from the stable $\infty$-category of motivic $\mathbb{P}^1$-spectra over $k$ to the stable $\infty$-category of $(H\mathbb{Z}/\ell)^*$-local pro-spectra (see section 3 for definition). This is induced by the étale topological realization functor á la Friedlander. The constant presheaf functor naturally induces the functor \[\text{SH}[1/p]\right…

Prismatic cohomology and $p$-adic homotopy theory

This short note regards an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the $p$-adic homotopy type for a smooth proper complex variety with good reduction mod $p$.

同伦自同构的表示稳定性

Representation stability for homotopy automorphisms

本文并行研究了拓扑空间迭代楔和与流形迭代连通和的同伦自同构，证明其有理同伦群构成有限生成FI-模，从而在Church和Farb的意义下满足对称群的表示稳定性。

球面纤维化在连通和上的拉回

Pullbacks of Sphere Fibrations over Connected Sums

研究球面纤维化在连通和上的拉回的同调等价条件，基于同调理论提出新方法，扩展至庞加莱对偶复形和局部情形。

Homotopy invariance of 4-manifold decompositions: connected sums

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.

连通和的环路空间分解及其在Vigué-Poirrier猜想中的应用

Loop Space Decompositions of Connected Sums and Applications to the Vigué-Poirrier Conjecture

本文扩展了分解高连通Poincaré对偶复形基点环路空间的方法，并应用于为有理同伦理论中长期存在的Vigué-Poirrier猜想提供新的支持实例。

具有有限基本群的正特征簇提升的一个必要条件

A necessary condition for liftings of positive characteristic varieties with finite fundamental groups

研究具有有限基本群的正特征簇提升到特征零的必要条件，基于平展同伦理论和沃尔有限性障碍，通过定义模l有限支配性概念，证明簇提升的链复形条件，还扩展了沃尔有限性障碍讨论。

Homotopy groups of highly connected manifolds

In this paper we give a formula for the homotopy groups of $(n-1)$-connected $2n$-manifolds as a direct sum of homotopy groups of spheres in the case the $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti number is $1$ the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore.

Motion planning in connected sums of real projective spaces

The topological complexity ${\sf TC}(X)$ is a homotopy invariant of a topological space $X$, motivated by robotics, and providing a measure of the navigational complexity of $X$. The topological complexity of a connected sum of real projective planes, that is, a high genus nonorientable surface, is known to be maximal. We use algebraic tools to show that the analogous result holds for connected sums of higher dimensional real projective spaces.

Homotopy Types of Suspended $4$-manifolds

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $Σ^2M$ as wedge sums of some elementary $\mathbf{A}_3^3$-complexes, which are $2$-connected finite complexes of dimension at most $6$. Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of $Σ^2M$ to desuspend to that of $ΣM$.

球面乘积连通和的同伦自等价的稳定上同调

The stable cohomology of self-equivalences of connected sums of products of spheres

本文确定了球面乘积连通和的同伦自等价稳定分类空间的上同调，并将其用李图复形同调表达。

Homotopy Theoretic Models of Type Theory

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it. On the other hand, those conditions are easy to check and provide a wide class of models some of which are listed in the paper.

Enumerating partitions arising in homotopy theory

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod $2$ Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory.

概形的幂幺同伦理论

Unipotent homotopy theory of schemes

本文基于 Toën 的仿射栈工作建立概形的幂幺同伦理论，证明其能恢复幂幺 Nori 基本群概形、$p$-进 étale 同伦群及 Artin-Mazur 形式群，并证明 Freudenthal 悬挂定理、有限性定理以及 Calabi-Yau 簇的导出不变量性质。

The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its étale version is …

L-无穷结构的同伦转移与BV形式体系

Homotopy transfer for L-infinity structures and the BV-formalism

本文利用BV形式体系和积分微扰展开，给出了广义和幺模L-无穷代数结构极小模型的显式构造，证明了其作为同伦转移定理的实例，并揭示了有根树公式与费曼图的关系。

Morphisms of rational motivic homotopy types

We investigate several interrelated foundational questions pertaining to the study of motivic dga's of Dan-Cohen--Schlank [8] and Iwanari [13]. In particular, we note that morphisms of motivic dga's can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga's is a certain limit of such spectra; we give an explicit formula for this limit --- a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids …

分支覆盖的一个同伦后果

A homotopical consequence of branched covers

研究证明伪流形的余有限完成等于其分支覆盖的Artin-Mazur层叠同伦类型构造，揭示了伪流形中足够多的K(π,1)稠密子空间的存在性。

Cech approximation to the Brown-Gersten spectral sequence

In this paper, we show that the etale index of a torsion cohomological Brauer class is divisible by the period of the class. The tool used to make this computation is the Cech approximation of the title. To create the approximation, we use the folklore theorem that the homotopy limit and Postnikov spectral sequences for a cosimplicial space agree beginning with the E2-page. As far we know, this folklore theorem has no proof in the literature, so we include a proof.

Motivic Atiyah-Segal completion theorem

Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebra…

Internal Languages of Finitely Complete $(\infty, 1)$-categories

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Löf Type Theory with dependent sums and intensional identity types is the internal language of $(\infty, 1)$-categories with finite limits.