Topological Complexity is a Fibrewise L-S Category

Topological complexity \TC{B} of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version \TCM{B} of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities \TC{B} = \catBb{\double{B}}+1 and \TCM{B} = \catBB{\double{B}}+1, where \double{B}=B{\times}B is a fibrewise pointed space over $B$ whose projection and section are given by p_{\double{B}}=\proj_{2} : B{\times}B \to B the canonical projection to the second factor and s_{\double{B}}=\Delta_{B} : B \to B{\times}B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. Recently, we found a problem with the proof of Theorem 1.13 which states that for a fibrewise well-pointed space X$over$B$, we have$\catBB{X} = \catBb{X} and that for a locally finite simplicial complex $B$, we have \TC{B} = \TCM{B}. While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants \catBb{X} and \catBB{X} is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12.Comment: 12pages original + 5pages errat

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