Implications of the Ganea Condition

利用弱拓扑复杂度研究拓扑复杂度的 Ganea 猜想

A study of the Ganea conjecture for topological complexity by using weak topological complexity

该研究借助 Berstein-Hilton 弱拓扑复杂度及其稳定形式，给出空间满足拓扑复杂度 Ganea 猜想的充分条件，并通过若干例子加以说明。

A_{\infty}-method in Lusternik-Schnirelmann category

To clarify the method behind the paper "Ganea's conjecture on Lusternik-Schnirelman category" by the author, a generalisation of Berstein-Hilton Hopf invariants is defined as `higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of "Ganea's ..." for all the cases except the case when …

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

A criterion to determine the L-S category of a total space of a sphere-bundle over a sphere is given in terms of homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for such a total space $N$ to have the same L-S category as its `once punctured submanifold' $N\smallsetminus\{P\}$, $P \in N$. Also a necessary condition for such a total space $M$ to satisfy Ganea's conjecture is obtained.

Ganea分解的分类空间

Ganea decompositions of classifying spaces

本文研究了通过纤维-纤维束构造获得的紧连通李群分类空间BG的Ganea分解，确定了纤维束的上同调条件，证明了分解的sharp性，并构造了多个例子。

Lusternik-Schnirelmann categories of non-simply connected compact simple Lie groups

Let $F \hookrightarrow X \to B$ be a fibre bundle with structure group $G$, where $B$ is $(d{-}1)$-connected and of finite dimension, $d \geq 1$. We prove that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim B}{d}$, if $F$ has a cone decomposition of length $m$ under a compatibility condition with the action of $G$ on $F$. This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain $\cat{PU(n)} \leq 3(n{-}1)$ for all $n \geq 1$, which might be best possible, since we have $\cat{\mathrm{PU}(p^r)}=3(p^r{-}1)$ f…