G-层伪流形的等变开折

Intersection Space Cohomology of Three-Strata Pseudomanifolds

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically fla…

叶分层空间与描述交空间上同调的De Rham复形

Foliated Stratified Spaces and a De Rham Complex Describing Intersection Space Cohomology

交空间的方法将依赖于反常的胞腔复形与某些类型的分层伪流形相关联，使得与互补反常相关联的胞腔复形的普通有理上同调群之间的庞加莱对偶成立。这些交空间的上同调定义了奇异空间的上同调理论HI，它与交上同调IH不同构。镜像对称倾向于互换IH和HI。IH理论可以与IIA型弦理论联系在一起，而HI理论可以与IIB理论联系在一起。对于层深为1且环丛平坦的伪流形，本文利用顶层上的整体光滑微分形式的复形给出了理论HI的de Rham-理论描述.证明了形式的楔积对每一反常都在H Ⅰ上引入一个反常内上积。例如，平坦链丛出现在叶状分层空间和局部对称空间的约化Borel-Serre紧化中。给出了分层叶理概念的精确拓扑定义。

The L-Homology Fundamental Class for IP-Spaces and the Stratified Novikov Conjecture

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric $L$-spectrum of $\Z$, which is, up to weak equivalence, an $E_\infty$ ring map. Using this map, we construct a fundamental $L$-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces.

Multiplicative de Rham Theorems for Relative and Intersection Space Cohomology

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincaré Duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth man…

伪流形的等变 L-类

The Equivariant L-Class of Pseudomanifolds

本文为紧致 Lie 群在满足 Witt 条件的分层紧致定向伪流形上的保向作用构造了等变 L-类，证明其性质并给出乘积公式，方法基于 Borel 构造的伪流形逼近与 Verdier-Riemann-Roch 型公式。

一般Peru的De Rham交上同调

De Rham intersection cohomology for general perversities

对于一个分层的伪流形$X$，我们有一个用于验证$\per{0} \leq \per{p} \leq \per{t}$的反常$\per{p}$的de Rham定理$ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} - \per{p}}{*}{X}, $，其中$\per{t}$表示顶部反常。我们把这个结果推广到任意反常的$\per{p}$。在方向上同调$\mapsto$上同调中，我们得到同构$$ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} -\per{p}}{*}{X,\ib{X}{\per{p}}}, $$，其中$ {\displaystyle \ib{X}{\per{p}} = \bigcup\_{S \preceq S\_{1} \atop \per{p} (S\_{1})< 0}S = \bigcup\_{\per{p} (S)< 0}\bar{S}.} $在方向同调$\mapsto$上同调中，我们得到同构$$ \lau{\IH}{\per{p}}{*}{X}=\lau{\IH}{*}{\max (\per{0},\per{t} -\per{p})}{X}. $$本文中允许一维层的分层伪流形.

Intersection Spaces, Equivariant Moore Approximation and the Signature

We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a suitable choice of structure group. As a by-product, we find new characteristic classes for fiber bundles admitting such approximations. For trivial bundles and flat bundles whose base has finite fundamental group these classes vanish. For oriented closed pseudomanifolds, we prove that the reduced rational cohomology of the intersection spaces satisfies globa…

关于分层去奇异化的泛函分析

On The Functorialrily Of Stratified Desingularizations

本文研究了微分形式下De Rham交上同调的光滑去奇异化问题。本文主要研究的是单空间范畴。我们构造了一个函子，它将每个M-Mather简单空间发送到一个称为其初等开折的光滑流形中。因此我们证明了准素开折是唯一的上Mather同构。

奇异空间的拓扑配边及其在分层L类上的应用

Topological Bordism of Singular Spaces and an Application to Stratified L-Classes

本文构造广义同调配边理论，为流形同伦分层集提供同胚不变几何基本类，并由此构建拓扑不变的L类，在pl-伪流形情形与Goresky-MacPherson L类一致。

光滑图册分层空间、K同调定向与Gysin映射

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps

该研究引入光滑图册分层空间并证明其与Thom-Mather分层空间等价，在双变KK-理论中构造定向Witt分层空间纤维丛的Gysin映射并证明其保持解析符号差类，还建立了解析符号差类与拓扑Siegel-Sullivan定向的联系。

呈现拓扑分层同伦假说

Presenting the topological stratified homotopy hypothesis

本文证明分层拓扑空间上存在呈现三种分层同伦理论的（半）模型结构，并利用Lurie函子建立Quillen等价，从而给出分层同伦假说的模型范畴解释。

Topological and Hodge L-Classes of Singular Covering Spaces and Varieties with Trivial Canonical Class

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We ap…

Witt空间的对称签名

The symmetric signature of a Witt space

Witt空间是伪流形，其中有理系数的中反常交同调是自对偶的。我们给出了一个新的结构的对称签名的维特空间，这是类似的精神，由Miscenko流形的建设。我们的构造具有所有预期的性质，包括分层同伦等价下的不变性。

传统和超permitting的奇异链相交同调

Singular chain intersection homology for traditional and super-perversities

我们引入了一个奇异链交同调论，它推广了King的理论，并与Goresky和MacPherson关于任何拓扑分层伪流形的Deligne层交同调一致，无论这些拓扑分层伪流形是紧的还是不紧的，具有常系数或局部系数，并且与传统的伪流形或超伪流形（满足p（2）>0的伪流形）一致.对于p（2）=1的情形，后一种反常性是由Cappell和Shaneson引入的，并且在他们的嵌入超对偶定理中起着关键作用。我们进一步描述了这个奇异链复形的层化及其对更广泛的分层空间类的适应性。