Clapp-Puppe Type Lusternik-Schnirelmann (Co)category in a Model Category

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological Complexity

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.

Mountain Pass Theorem With infinite symmetry

We extend a work of Bartsch, Clapp and Puppe on the Mountain pass theorems. We consider functionals invariant with respect to infinite discrete groups satisfying a maximality condition on the finite subgroups.

Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(π)$ denotes the cohomological dimension of a group $π$ and ${\rm hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: \[ {\rm cat} X\leq \Bigl\lceil\frac{{\rm hd} (X)}{k+1}\Bigr\rceil \text{ if } π_i(X)=0 \text{ for } i\leq k. \] We make a synthesis and generalization of both of these results, by demonstrating the main result: \[ {\rm cat}\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{k+1}\…

Ganea和Whitehead定义的叶状流形的切向Lusternik-Schnirelmann范畴

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

本文在分层空间范畴中发展了Ganea和Whitehead定义的叶状流形切向Lusternik-Schnirelmann范畴，通过比较与开集定义的第三不变量，证明在闭流形上这些定义与切向范畴一致且为同调不变量。