59 research outputs found
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, -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 along the classifying map of the universal covering. If such
a space X is actually a co-H-space, then the fibrewise p-localization of
(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
along , 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 form on a diffeological loop space
To construct an -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 -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 -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 be a fibre bundle with structure group ,
where is -connected and of finite dimension, . We prove
that the strong L-S category of is less than or equal to , if has a cone decomposition of length under a compatibility
condition with the action of on . 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 , which might be best
possible, since we have \cat{\mathrm{PU}(p^r)}=3(p^r{-}1) for any prime
and . Similarly, we obtain the L-S category of for
and . 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
We give a cellular decomposition of the compact connected Lie group
. We also determine the L-S categories of and .Comment: 14 page
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