arXiv subjects

Robert F. Brown

Publications and source records attributed to Robert F. Brown.

3 recordsLinked to original sources

arXiv2019

Lifting classes for the fixed point theory of $n$-valued maps

The theory of lifting classes and the Reidemeister number of single-valued maps of a finite polyhedron $X$ is extended to $n$-valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicoténcatl. The liftings of an $n$-valued map $f$ split into self-maps of the universal covering space of $X$ that we call lift-factors. An equivalence relation is defined on the lift-factors of $f$ and the number of equivalence classes is the Reidemeister number of $f$. The fixed point classes of $f$ are the projections of the fixed point sets of the lift-factors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of $X$ such that the number of equivalence classes equals the Reidemeister number. We prove that if $X$ is a manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to $n$-valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an $n$-valued map equals its Reidemeister number. If an $n$-valued map splits into $n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of their Reidemeister numbers.

math.AT↗

arXiv2004

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.

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arXiv2004

The boundary-Wecken classification of surfaces

Let X be a compact 2-manifold with nonempty boundary dX and let f: (X, dX) --> (X, dX) be a boundary-preserving map. Denote by MF_d[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N_d(f) is the sum of the number of essential fixed point classes of the restriction f-bar : dX --> dX and the number of essential fixed point classes of f that do not contain essential fixed point classes of f-bar. We prove that if X is the Moebius band with one (open) disc removed, then MF_d[f] - N_d(f) < 2 for all maps f : (X, dX) --> (X, dX). This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If X is the disc, annulus or Moebius band, then X is boundary-Wecken, that is, MF_d[f] = N_d(f) for all boundary-preserving maps. If X is the disc with two discs removed or the Moebius band with one disc removed, then X is not boundary-Wecken, but MF_d[f] - N_d(f) < 2. All other surfaces are totally non-boundary-Wecken, that is, given an integer k > 0, there is a map $f_k : (X, dX) --> (X, dX) such that MF_d[f_k] - N_d(f_k) >= k.

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