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Coincidences of self-maps on Klein bottle fiber bundles over the circle. (English) Zbl 1246.55001
Consider fiber preserving maps \(f_1,f_2: E\to E\) where \(E\) is a fiber bundle over \(S^1\) and the fiber is the Klein bottle \(K\). Then the fiber bundle \(K@>>>E@>p>>S^1\) is of the form \(K@>>>M(\phi)@>p>>S^1\) for a homeomorphism \(\phi\) of \(K\) and the total space \(M(\phi)\) is the quotient of \(K\times I\) by the relation \((x,0)\sim(\phi(x),1)\). Denote by \(M(\phi)\times_{S^1}M(\phi)\) the pullback of \(p:M(\phi)\to S^1\). Let \(E_{S^1}(M(\phi))=\{(x,\omega)\in (M(\phi)\times_{S^1}M(\phi)\setminus\Delta)\times (M(\phi)\times_{S^1}M(\phi))^{I}|\;x=\omega(0)\}\). Then there is a fiber bundle \(q:E_{S^1}(M(\phi))\to M(\phi)\times_{S^1}M(\phi)\) where \(q(x,\omega)=\omega(1)\).
The authors study the question of when a pair of fiber-preserving maps \((f_1,f_2):M(\phi)\to M(\phi)\) can be deformed by a fiberwise homotopy over \(S^1\) to a coincidence free pair. They prove that this is possible iff there is a homomorphism \(\psi:\pi_1(M,\phi)\to\pi_1(E(M(\phi)\times_{S^1}M(\phi)\setminus\Delta))\) such that \(q_\#\circ\psi=(f_1,f_2)_\#\). Moreover, the authors explicitly list all homotopy classes of pairs \((f_1,f_2)\) such that \((f_1|\,K,f_2|\,K)\) can be deformed into a coincidence free pair. In addition, the authors derive necessary and sufficient conditions for the existence of the homomorphism \(\psi\) above. In reading this paper the reader will have to work through extensive calculations. It would be advantageous to read the articles [Fundam. Math. 183, No. 1, 1–38 (2004; Zbl 1060.55001), and ibid. 203, No. 3, 263–292 (2009; Zbl. 1167.55001)] by D. L. Gonçalves, D. Penteado and J. P. Vieira before indulging in the present article.

MSC:
55M20 Fixed points and coincidences in algebraic topology
55R10 Fiber bundles in algebraic topology
55S35 Obstruction theory in algebraic topology
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