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Fixed points on trivial surface bundles over a connected CW-complex. (English) Zbl 1374.55001
The main topic in fixed point theory is the existence of fixed points of self-maps. From the algebraic point of view, one may ask if these fixed points are stable under homotopies. In other words, can we make a self-map fixed point free by a homotopy? The authors of this paper consider this question in the case of fibre-preserving self-maps on fibre bundles. The case of a special trivial bundle $$S^1\times S_2$$ was already known. Here, $$S_2$$ means the closed orientable surface of genus $$2$$.
The present work solves the question for trivial bundles of the form $$B\times S$$, where the base $$B$$ is a connected CW-complex and $$S$$ is a closed surface of negative Euler characteristic. The authors consider fibrewise maps of the form $$h(x,y) = (x, f(x,y))$$. The main result shows that such an $$h$$ can be fibrewisely deformed into a fixed point free map if and only if $$h$$ is fibrewisely homotopic to $$id\times g$$, where $$g: S\to S$$ is a fixed point free self-map homotopic to the given self-map $$f: x_0\times S\to x_0\times S$$.
##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 55R10 Fiber bundles in algebraic topology
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