Periodic point free continuous self-maps on graphs and surfaces

AbstractWe prove the following three results. We denote by Per(f) the set of all periods of a self-map f.Let G be a connected compact graph such that dimQH1(G,Q)=r, and let f:G→G be a continuous map. If Per(f)=∅, then the eigenvalues of f⁎1 are 1 and 0, this last with multiplicity r−1, where f⁎1 is the induced action of f on the first homological space.Let Mg,b be an orientable connected compact surface of genus g⩾0 with b⩾0 boundary components, and let f:Mg,b→Mg,b be a continuous map. The degree of f is d if b=0. If Per(f)=∅, then the eigenvalues of f⁎1 are 1, d and 0, this last with multiplicity 2g−2 if b=0; and 1 and 0, this last with multiplicity 2g+b−2 if b>0.Let Ng,b be a non-orientable connected compact surface of genus g⩾1 with b⩾0 boundary components, and let f:Ng,b→Ng,b be a continuous map. If Per(f)=∅, then the eigenvalues of f⁎1 are 1 and 0, this last with multiplicity g+b−2.The tools used for proving these results can be applied for studying the periodic point free continuous self-maps of many other compact absolute neighborhood retract spaces