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We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=\textbf{Diff}$ or $\textbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $\mathbb{R}^{n}$ decidable? We show that PL immersibility is decidable in all cases except for codimension 2, whereas smooth immersibility is decidable in all odd codimensions and undecidable in many even codimensions. As a corollary, we show that the smooth embeddability of an $m$-manifold with boundary in $\mathbb{R}^{n}$ is undecidable when $n-m$ is even and $11m \geq 10n+1$.

 

For a finite group$G$of not prime power order, Oliver showed that the obstruction for a finite CW-complex$F$to be the fixed point set of a contractible finite$G$-CW-complex is determined by the Euler characteristic$\chi (F)$. (He also has similar results for compact Lie group actions.) We show that the analogous problem for$F$to be the fixed point set of a finite$G$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on$K_0$[2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

 

Smith theory says that the fixed point set of a semi-free action of a group$G$on a contractible space is${\mathbb {Z}}_p$-acyclic for any prime factor$p$of the order of$G$. Jones proved the converse of Smith theory for the case$G$is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain$K$-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of$K$-theoretical obstructions.

 

Parametrized motion planning algorithms \cite{CFW} have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles.Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.