余H空间中Ganea猜想的p-完全形式

球面上球面丛的Lusternik-Schnirelmann范畴

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

利用球面上球面丛的特征映射的同伦不变量，给出了判定球面上球面丛全空间的L-S范畴的一个判据，从而完整地回答了Ganea问题4.结果得到了这样的全空间$N$与它的“一次穿孔子流形”$N\smallsetminus\{P\}$，$P \in N$具有相同的L-S范畴的一个充要条件.并得到了这样一个全空间$M$满足Ganea猜想的必要条件.

Generating functions and topological complexity

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the pole equals $\cat(X)$. Here $X$ is a finite CW-complex and for $r\ge 2$ the symbol $\tc_r(X)$ denotes its $r$-th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.

On Iwase's manifolds

In ~\cite{Iw2} Iwase has constructed two 16-dimensional manifolds $M_2$ and $M_3$ with LS-category 3 which are counter-examples to Ganea's conjecture: ${\rm cat_{LS}} (M\times S^n)={\rm cat_{LS}} M+1$. We show that the manifold $M_3$ is a counter-example to the logarithmic law for the LS-category of the square of a manifold: ${\rm cat_{LS}}(M\times M)=2{\rm cat_{LS}} M$. Also, we construct a map of degree one $$f:N\to M_2\times M_3$$ which reduces Rudyak's conjecture to the question whether ${\rm cat_{LS}}(M_2\times M_3)\ge 5$. We show that ${\rm cat_{LS}}(M_2\times M_3)\ge 4$.

基于伴随和LS余范畴的Goodwillie演算

Goodwillie Calculus via Adjunction and LS Cocategory

本文证明了对空间F的约化同伦内函子和对所有的$n \geq 1$，存在伴随函子$R_n, L_n$与$T_n F \simeq R_n F L_n$，其中$P_n F$是$F$的$n$ -消去逼近，它是通过取$T_n F$的迭代上的同伦余极限而构造的.这就赋予了单位元的$T_n$一个单子的结构，而$T_n F$是该单子上双模的函子版本。因此，每个$T_n F$（和$P_nF$）取值于对称的Lusternik-Schnirelman余范畴$n$的空间，如霍普金斯所定义的。这也恢复了Chorny-Scherer最近的结果。空间$T_n F(X)$实际上是经典幂零的（在Berstein-Ganea意义下），但不是Biedermann和Dwyer意义下的幂零。我们将对偶微积分的原始构造扩展到我们的环境中，建立了函子的$n$ -共除近似，并将我们的构造对偶化，以获得关于构造$T^n$、$P^n$和LS范畴的类似结果。

Deligne猜想的瑞士奶酪版证明

Proof of Swiss Cheese Version of Deligne's Conjecture

对于一个结合代数A，我们考虑“Hochschild上链复形C*（A，A）和代数A”这对.有一个自然的2色操作数作用于这对。我们表明，这个操作是准同构的奇异链操作的Voronov的瑞士奶酪操作。这个陈述是M. Kontsevich in arXiv：math/9904055.

Monoidal结构的单纯局部化和Deligne猜想的非线性版本

Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture

证明了如果$(M,\tensor,I)$是么半群模型范畴，则$\REnd_M(I)$是$\sSet$中的（弱）2-么半群.当$M$是单纯幺半群$A$上的$A$ -双模的范畴时，这一点尤其适用：$A$的导出自同态形成了它的Hochschild上同调，从而成为单纯2-幺半群。

Regular polygraphs and the Simpson conjecture

We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two different notions of "non-unital categories". The "general" version involve infinity-categories where absolutely all composition operations (horizontal, vertical and whiskering) are defined and compatible, the "regular" version involve infinity-categories where all the composition operations corresponding to "regular" pasting diagram are defined and compatibl…

A counterexample to a strong version of the Andrews-Curtis conjecture

We prove that the presentations $\langle x,y | [x,y],1 \rangle$ and $\langle x,y | [x,[x,y^{-1}]]^2y[y^{-1},x]y^{-1},[x,[[y^{-1},x],x]] \rangle$ are not $Q^*$-equivalent even though their standard complexes have the same simple homotopy type.

母题亚当斯猜想

The motivic Adams conjecture

本文在反转基域指数特征的条件下证明了母题亚当斯猜想，并由此建立了模 k 多尔德定理与布朗技巧的母题版本，同时证明更高母题稳定同伦群具有有界挠。