Periodic point free continuous self-maps on graphs and surfaces

We prove the following three results. We denote by Per ( f ) the set of all periods of a self-map f. be a connected compact graph such that dim Q H 1 ( 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. g , b be an orientable connected compact surface of genus g ⩾ 0 with b ⩾ 0 boundary components, and let f : M g , b → M g , 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 2 g − 2 if b = 0 ; and 1 and 0, this last with multiplicity 2 g + b − 2 if b > 0 . g , b be a non-orientable connected compact surface of genus g ⩾ 1 with b ⩾ 0 boundary components, and let f : N g , b → N g , 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 . ols 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.