Parametrized Borsuk-Ulam problem for projective space bundles

Let $\pi: E \to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $\pi^{'}: E^{'} \to B$ be vector bundle such that $\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $\pi^{'}:E^{'} \to B$. For a fiber preserving $\mathbb{Z}_2$-equivariant map $f:E \to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \{x \in E | f(x)= 0\}.$ As an application, we also estimate the cohomological dimension of the $\mathbb{Z}_2$-coincidence set $A_f=\{x \in E | f(x) = f(T(x)) \}$ of a fiber preserving map $f:E \to E^{'}$.Comment: 14 pages, to appear in Fundamenta Mathematica

Z_2 actions on complexes with three non-trivial cells

In this paper, we study $Z_2$ actions on a cell complex X having the cohomology ring isomorphic to that of the wedge sum $P^2 (n) V S^{3n}$ or $S^n V S^{2n} V S^{3n}$. We determine the possible fixed point sets depending on whether or not X is totally non-homologous to zero in $X_{Z_2}$ and give examples realizing the possible cases.Comment: 7 pages, appeared in Topology and its Application

A note on the commutativity of inverse limit and orbit map

We show that the inverse limit and the orbit map commute for actions of compact groups on compact Hausdorff spaces.Comment: 4 page