Reidemeister classes for coincidences between sections of a fiber bundle

Resumo

Let $s_0,f_0$ be two sections of a fiber bundle $q: E\to B$ and the coincidence set $\Gamma(s_0,f_0)\neq \emptyset$. We consider the following question:  Is there $s_0\simeq_B s_1$ (by the homotopies which cover the constant homotopy $\overline{I}_B$ on the basic space) such that $\Gamma(s_1, f_0)=\emptyset$? If $b_0\in \Gamma(s_0,f_0)$ and $F_0=q^{-1}(b_0)$ is the typical fiber, in this context we can use the homotopy lifting extension propriety of the fibration $q$ to obtain homotopies over $B$. When we make this and the basic point are fixed we can use the elements $s_0(\beta), f_0(\beta^{-1})$ where $\beta \in\pi_1(B,b_0)$ and the elements $\gamma\in \pi_1(F_0,e_0)$. So we will introduce the algebraic classes of Reidemeister relative to the subgroup $\pi_1(F_0,e_0)$. When the basic points are not fixed we need to consider the classes $[\til{s}]_L$ of lifting of $s_0$ defined on the universal covering $\til{B}$ to $\til{E}$. The present work relates the lifting classes $[\til{s}]_L$ of $s_0$ and the algebraic relative Reidemeister classes $R_A(s_0,f_0; \pi_1(F_0,e_0).$