Coincidence of pairs of maps on torus fibre bundles over the circle

Abstract

Let f, g: M(ϕ1) → M(ϕ2) be fibre-preserving maps over the circle, S1, where M(ϕ1) and M(ϕ2) are fibre bundles over S1 and the fibre is the torus, T. The main purpose of this work is to classify the pairs of maps (f, g) which can be deformed by fibrewise homotopy over S1 to a coincidence-free pair (f′, g′) , f′, g′: M(ϕ1) → M(ϕ2). In general, the classification of such pairs of maps is equivalent to finding solutions for an equation in the free group π2(T, T- 1) , called the main equation. In certain situations, it is appropriate to study the main equation in the abelianization of π2(T, T- 1) or on some quotients of this group, since, if the equation in one of these quotients does not admit solution, then the original equation also does not admit solution. In this case, it is not possible to obtain the desired deformability.