Strategies to annihilate coincidences of maps from two-complexes into the circle (Q2090633)

Let \(f,g:k\to S^{1}\) be two maps from a finite and connected two-dimensional CW complex, with essential coincidence points. Considering the inclusions \(\iota: S^{1}\to S^{1}\vee [0,1]\), \(\varepsilon :S^{1}\to S^{1}\vee S^{1}\) and \(\tau:S^{1}\to \Theta\), where \(\Theta\) is the space obained by gluing the two endpoints of \([0,1]\) to two different points of \(S^{1}\), it is shown that if one of the pairs of maps \((\iota\circ f,\iota\circ g)\), \((\varepsilon\circ f,\varepsilon\circ g)\) and \((\tau\circ f,\tau\circ g)\) can be deformed to be coincidence free, so can the other ones. Furthermore, some results concerning the coincidence problem for maps from certain classes of closed surfaces into \(S^{1}\vee S^{1}\) and \(\Theta\) are established. More precisely, if \(X\neq S^{1},\mathbb{R} P^{2}\), then there is a strongly surjective map \(X\to S^{1}\vee S^{1}\) and, in general, each pair of maps from the Klein bottle into \(S^{1}\vee S^{1}\) can be deformed to be coincidence free, with the latter one being a consequence of the following: Let \(\mathbb{T}_{k}\) and \(\mathbb{U}_{k}\) be the orientable and the non-orientable closed surface of genus \(k\geq 1\), respectively. If \(\text{im}(f,g)_{\sharp}\) is cyclic then a pair of based maps \(f,g:X\to S^{1}\vee S^{1}\) can be deformed to be coincidence free if \(X=\mathbb{T}_{k}\), with \(k\geq 1\), or if \(X=\mathbb{U}_{k}\), with \(k\geq 2\).