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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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