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Coincidence points of fiber maps on $$S^n$$-bundles. (English) Zbl 1196.55001
The authors study coincidences for fiber-preserving maps between $$S^n$$-bundles. If $$E\to B$$ is an $$S^n$$-bundle they first show the existence of a fiber-preserving map $$\tau:E\to E$$ which is fixed point free and such that the degree of $$\tau|\,S^n$$ is 1 if $$n$$ is odd and $$-1$$ if $$n$$ is even. They then obtain the following result: Let $$E_1\to B$$ and $$E_2\to B$$ be $$S^n$$-bundles and $$f:E_1\to E_2$$ a fiber-preserving map. Then there is a unique homotopy class of maps given by $$\mathcal{C}=[\tau\circ f]$$ such that for $$g\in\mathcal{C}(f)$$ the pair $$(f,g)$$ can be deformed over $$B$$ to a coincidence free pair. The authors then study the case of $$S^n$$-bundles over $$S^1$$.

##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 55R10 Fiber bundles in algebraic topology
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##### References:
 [1] Brooks, R., On removing coincidences of two maps when only one, rather than both of them, may be deformed by a homotopy, Pacific J. math., 40, 1, 45-52, (1972) · Zbl 0235.55006 [2] Brown, R.F., The Nielsen number of a fibre map, Ann. of math., 85, 2, 483-493, (1967) · Zbl 0149.20304 [3] Brown, R.F., Fixed points and fibre, Pacific J. math., 21, 465-472, (1967) · Zbl 0169.25801 [4] Dold, A., The fixed point index of fibre-preserving maps, Invent. math., 25, 281-297, (1974) · Zbl 0284.55007 [5] Fadell, E.; Husseini, S., A fixed point theory for fiber preserving maps, Lecture notes in mathematics, vol. 886, (1981), Springer-Verlag, pp. 49-72 · Zbl 0485.55002 [6] Ferrario, D., Computing Reidemeister classes, Fund. math., 158, 1, 1-18, (1998) · Zbl 0915.55001 [7] Gonçalves, D.L., Fixed point of $$S^1$$-fibrations, Pacific J. math., 129, 297-306, (1987) · Zbl 0608.55003 [8] Gonçalves, D.L.; Koschorke, U., Nielsen coincidence theory of fibre-preserving maps and Dold’s fixed point index, Topol. methods nonlinear anal., 33, 1, 85-103, (2009) · Zbl 1178.55002 [9] Gonçalves, D.L.; Penteado, D.; Vieira, J.P., Fixed points on torus fiber bundles over the circle, Fund. math., 183, 1, 1-38, (2004) · Zbl 1060.55001 [10] Gonçalves, D.L.; Penteado, D.; Vieira, J.P., Fixed points on Klein bottle fiber bundles over the circle, Fund. math., 203, 3, 263-292, (2009) · Zbl 1167.55001 [11] Gonçalves, D.L.; Penteado, D.; Vieira, J.P., Abelianized obstruction for fixed points of fiber-preserving maps of surface bundles, Topol. methods nonlinear anal., 33, 2, 293-305, (2009) · Zbl 1195.55003 [12] Gonçalves, D.L.; Randall, D., Self-coincidence of maps from $$S^q$$-bundles over $$S^n$$ to $$S^n$$, Bol. soc. mat. mexicana, 10, 3, 181-192, (2004), (Special Issue) · Zbl 1097.55001 [13] Hansen, V.L., On the space of maps of a closed surface into the 2-sphere, Math. scand., 35, 149-158, (1974) [14] Toda, H., Composition methods in homotopy groups of spheres, Annals of mathematics studies, vol. 49, (1962), Princeton University Press · Zbl 0101.40703 [15] Stasheff, J.D., A classification theorem for fibre spaces, Topology, 2, 239-246, (1963) · Zbl 0123.39705 [16] Whitehead, G., Elements of homotopy theory, (1978), Springer-Verlag New York · Zbl 0406.55001
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