Splitting off Rational Parts in Homotopy Types

It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (See Theorem 21.3 of \cite{Fuchs:abelian-group}). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say \bar{\rho} : [S_{\Q}^{n},X] \to H_n(X;\Z) and generalized Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, $T$-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.Comment: 6 page

Co-H-spaces and almost localization

Apart from simply-connected spaces, a non simply-connected co-H-space is a typical example of a space X with a co-action of $B\pi_1(X)$ along $r^X : X \rightarrow B\pi_{1}(X)$ the classifying map of the universal covering. If such a space X is actually a co-H-space, then the fibrewise p-localization of $r^X$ (or the `almost' p-localization of X) is a fibrewise co-H-space (or an `almost' co-H-space, resp.) for every prime p. In this paper, we show that the converse statement is true, i.e., for a non simply-connected space X with a co-action of $B\pi_1(X)$ along $r^X$, X is a co-H-space if, for every prime p, the almost p-localization of X is an almost co-H-space.Comment: 10 pages, no figure

Smooth A∞A_{\infty} form on a diffeological loop space

To construct an $A_{\infty}$-form for a loop space in the category of diffeological spaces, we have two minor problems. Firstly, the concatenation of paths in the category of diffeological spaces needs a small technical trick (see P.~I-Zemmour \cite{MR3025051}), which apparently restricts the number of iterations of concatenations. Secondly, we do not know a natural smooth decomposition of an associahedron as a simplicial or a cubical complex. To resolve these difficulties, we introduce a notion of a $q$-cubic set which enjoys good properties on dimensions and representabilities, and show, using it, that the smooth loop space of a reflexive diffeological space is a h-unital smooth $A_{\infty}$-space. In appendix, we show an alternative solution by modifying the concatenation to be stable without assuming reflexivity for spaces nor stability for paths.Comment: 18 page

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) for any prime $p$ and $r \geq 1$. Similarly, we obtain the L-S category of $\mathrm{SO}(n)$ for $n \leq 9$ and $\mathrm{PO}(8)$. We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.Comment: 13 page

On the cellular decomposition and the Lusternik-Schnirelmann category of Spin(7)Spin(7)

We give a cellular decomposition of the compact connected Lie group $Spin(7)$. We also determine the L-S categories of $Spin(7)$ and $Spin(8)$.Comment: 14 page