Propagating sharp group homology decompositions

Homology of FI-modules

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.

谱最小分割的同调

Homology of spectral minimal partitions

本文通过定义修正拉普拉斯算子并借助代数拓扑中的相对同调工具，研究了非二分谱最小分割，证明其 Courant-sharp 特征函数的节点分割在特定拓扑类中最小，同时统一恢复了二分情形的已知结果。

Some Computations on Instanton Knot Homology

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard Diagrams of 3-manifolds. For a knot inside S3, we further develop an algorithm that can compute an upper bound of the dimension of instanton knot homology from knot diagrams. We test the effectiveness of the algorithm and found that for all knots up to seven crossings, the algorithm provides sharp bounds. In the second half of the paper, we show that, if the instanton knot Floer homology of a knot has a specified form, then the knot mus…

Tanabe代数的上同调

Cohomology of Tanabe algebras

本文证明Tanabe代数的（上）同调与对称群的（上）同调同构，且与参数和索引奇偶性无关，首次展示此类结果。

通过西尔皮ński四面体的切片的拓扑

Topology of slices through the Sierpiński tetrahedron

该研究探讨了西尔皮ński四面体在不同高度处的切片拓扑结构，揭示了二进制有理数与非二进制有理数对切片连通性和同调群的影响。

Topological K-(co-)homology of classifying spaces of discrete groups

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology of the classifying space BG up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculat…

构型空间的乘性Chow-Künneth分解与同调分裂

Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces

本文构造了光滑固有簇点构型空间上同调的分裂，证明了超椭圆Torelli群作用平凡及Leray谱序列立即退化，并据此对$M_{2,n}$和$H_{g,n}$的上同调进行了详细计算。

特征二下的卡尔森猜想与广义总秩猜想

Carlsson's Conjecture and the Generalized Total Rank Conjecture in Characteristic Two

研究特征2正则环上广义总秩猜想，证明了初等阿贝尔2-群卡尔森猜想，通过识别链模型同调与弗罗贝尼乌斯拉回、变形泰特微分比较长度，得出相关同调界及球面秩猜想。

Vanishing theorems for representation homology and the derived cotangent complex

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := π_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, l…

Temperley-Lieb 代数的同调

The homology of the Temperley-Lieb algebras

本文借鉴群族同调稳定性方法，构造平面内射词链复形与归纳分解，证明 TL_n(a) 的同调和上同调在次数 n-2 以内消失，并确定偶数 n 时范围最佳。

Mapping class groups of highly connected $(4k+2)$-manifolds

We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.