分层伪流形上$S^1$ -作用的Gysin序列

一般的Permian定理与L^2 de Rham和Hodge定理

General perversities and L^2 de Rham and Hodge theorems for stratified pseudomanifolds

给出了一个紧致的分层伪流形，其正则部分上具有一个E-Mather分层和一类黎曼度量，研究了$L^{2}$的de Rham和Hodge上同调与$X$的交上同调之间的关系.更确切地说，对于我们称之为\emph{quasi edge with weights}的一类度量，我们将G意义下的两个一般常数联系起来。Friedman，$p_{g}$及其对偶$q_{g}$ .然后，我们表明，绝对$L^{2}$霍奇上同调同构的最大$L^{2}$德拉姆上同调，这又是同构的交叉上同调与反常$q_{g}$。证明了相对$L^{2}$ Hodge上同调同构于极小$L^{2}$ de Rham上同调，而极小的$L^{2}$又同构于反常$p_{g}$的交上同调.

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。

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…

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.

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。

层状纤维化与底层正则邻域的交同调

Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata

本文利用Leray-Serre型谱序列计算了分层PL-伪流形底层的正则邻域和消去正则邻域的交同调。谱序列的E^2项由底层与局部系数系统的同调给出，局部系数系统的柄由该层的链环（或该链环上的锥）的交集同调模组成。在这个程序中，我们建立了非滤子底空间上的分层纤维化及其映射柱的性质。我们还证明了一个关于相交同调的层保持同伦不变性的民间定理。

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…

Blown-up intersection cohomology

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivan's minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon on Steenrod squares in intersection homology. In this paper, we establish the main properties of this cohomology. One of its major feature is the existence of cap and cup products for any filtered space and any commutative ring of coefficients, at the cochain level. Moreover, we show that each stratified map induces an homomorphism between the blown-up inte…