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Coincidence of pairs of maps on torus fibre bundles over the circle. (English) Zbl 1437.55004
Denote by $$M(\Phi )$$ a fibre bundle over the circle $$S^1$$, where the fiber is the torus $$T$$. This space is obtained from $$T\times [0,1]$$ by identifying the points $$(x,0)$$ with the points $$(\Phi(x),1)$$, where $$\Phi:T\to T$$ is a homeomorphism. The authors consider fibre-preserving maps $$f,g:M(\Phi_1)\to M(\Phi_2)$$ over $$S^1$$. The purpose of the paper is to classify the pairs of maps $$(f,g)$$ which can be deformed by fibre-preserving homotopy to a coincidence-free pair $$(f',g')$$. This problem is equivalent to finding solutions of some equation in the free group $$\pi _2(T,T-1)$$. This equation is carefully investigated. The special case with $$\Phi_1=\Phi_2$$ was considered by the first author in [Topol. Methods Nonlinear Anal. 46, No. 2, 507–548 (2015; Zbl 1366.55002)].
##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 55R10 Fiber bundles in algebraic topology
##### Keywords:
coincidence; $$T$$-fibre bundles; fibre-preserving maps
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##### References:
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