Topological complexity and L-S theory

酒井酒井和岩濑井展示了以下内容。1)当且仅当X是C_k空间时，一个cat(X)小于等于k的空间X是h空间。此外，除基点外，C_1空间最多有三个单元格是h空间。2)一元拓扑复杂度与原始拓扑复杂度最多相差1。3)旋转群的L-S类与其杯长重合。4) L-S范畴小于或等于1当且仅当L-S范畴的p-局域化版本对每个素数p具有相同的性质，其中我们不需要假设空间是单连通的。在此基础上，Sumi et al进一步给出了有限群G是间隙群的充分必要条件，除非第一个同调群H_1(G)是2群，并证明了Laitinen猜想对除Aut（A_6）外的任何有限不可解群都成立。

Sakai and Iwase showed the folloing. 1) A space X with cat(X) less than or equal to k is an H-space if and only if X is a C_k soace. Moreover, a C_1 space with at most three cells other than the base point is an H-space. 2) A monoidal topological complexity differs from the original topological complexity at most by 1. 3) The L-S category of a rotation group coincides with its cup-length. 4) L-S category is less than or equal to 1 if and only if p-localized version of L-S category has the same property for every prime p, where we do not need to assume that the space is simply-connected. Further Sumi et al gave 5) a necessary and sufficient condition for a finite group G to be a gap group, unless the first homology group H_1(G) is a 2-group and showed 6) The Laitinen conjecture is true for any finite non-solvable groups except for Aut(A_6).