Let \(f,g: M(\phi _1)\rightarrow M(\phi _2)\) be fibre-preserving maps over the circle, \(S^1\), where \(M(\phi _1)\) and \(M(\phi _2)\) are fibre bundles over \(S^1\) 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 \(S^1\) to a coincidence-free pair \((f^{\prime },g^{\prime })\), \(f^{\prime },g^{\prime }: M(\phi _1)\rightarrow M(\phi _2)\). In general, the classification of such pairs of maps is equivalent to finding solutions for an equation in the free group \(\pi _2(T,T-1)\), called the main equation. In certain situations, it is appropriate to study the main equation in the abelianization of \(\pi _2(T, T-1)\) 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.

中文翻译：

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圆环上的环形纤维束上的成对图重合

令\（f，g：M（\ phi _1）\ rightarrow M（\ phi _2）\）是圆（\（S ^ 1 \））上的保纤映射，其中\（M（\ phi _1）\ ）和\（M（\披_2）\）是纤维束在\（S ^ 1 \）和所述纤维是环面，Ť。这项工作的主要目的是将可通过\（S ^ 1 \）上的纤维同向变形的图对（f，  g）分类为无巧合对\（（f ^ {\ prime}，g ^ {\ prime}）\），\（f ^ {\ prime}，g ^ {\ prime}：M（\ phi _1）\ rightarrow M（\ phi _2）\）。通常，此类映射对的分类等同于在自由组\（\ pi _2（T，T-1）\）中找到一个方程，该方程称为主方程。在某些情况下，应该以\（\ pi _2（T，T-1）\）的阿贝尔化或该组的某些商研究主方程，因为如果这些商之一中的方程不接受解，那么原始方程式也不接受解。在这种情况下，不可能获得所需的变形性。