https://www.degruyter.com/view/j/crll.2000.2000.issue-524/crll.2000.056/crll.2000.056.pdf

Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds

Nilmanifolds are Jiang-type spaces for coincidences

Let G be a compact connected Lie group, K a closed subgroup (not necessarily connected) and M = G/K the homogeneous space of left cosets. Assume that M is orientable and p * : H n ( G ) → H n ( M ) is nonzero, where n = dim M . In this paper, we employ an equivariant version of Nielsen root theory to show that the converse of the Lefschetz fixed-point theorem holds true for all selfmaps on M . Moreover, if the Lefschetz number of a selfmap f : M → M is nonzero, then the Nielsen number of f coincides with the Reidemeister number of f , which can be computed algebraically.

Fixed-point theory for homogeneous spaces

Let \(f \colon S_h \to S_g\) be a continuous mapping between orientable closed surfaces of genus h and g and let c denote the constant map \(c \colon S_h \to S_g\) with \(c(S_h) = c\in S_g\). Let \(\varrho(f)\) be the minimal number of roots of f' among all maps f' homotopic to f, i.e. \(\varrho(f) = \min \{|f'^{-1}(c)| : f' \simeq f \colon S_h \to S_g \}\). We prove that

The minimal number of roots of surface mappings and quadratic equations in free groups

Let f:M1→M2 be a continuous map and c:M1→M2 a constant map between closed (not necessarily orientable) surfaces. By definition the pair (f,c) has the Wecken property if f can be deformed into a map f' such that the number of coincidence points of (f',c) is the same as the number of essential coincidence classes of (f,c) and, hence, every essential coincidence class consists of exactly one point. When both surfaces are orientable the problem to determine all maps which have the Wecken property was solved in [14]. Let A(f) denote the absolute degree as defined in [6] or [15] and \(\). Here we show that a map f has the Wecken property iff either the Euler characteristic \(\) or \(\). In free groups there are solved certain quadratic equations closely related to the root problem.

Roots of mappings on nonorientable surfaces and equations in free groups

Certain Subgroups of the Fundamental Group and the Number of Roots of f(x) = a

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Certain Subgroups of the Fundamental Group and the Number of Roots of f(x) = a