Fixed points on trivial surface bundles over a connected CW-complex.

*(English)*Zbl 1374.55001The 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 \(f: x_0\times S\to x_0\times S\).

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 \(f: x_0\times S\to x_0\times S\).

Reviewer: Xuezhi Zhao (Beijing)