| Let s : N → Z be an integer sequence. To this sequence we associate the Mobius inverse sequence, denoted Ms : N → Z , which is defined as follows: Msn= d∣nmd snd Let X be a Euclidean Neighborhood Retract (ENR) and f : X → X a continuous map. Denote by Lambda(f) the Lefschetz number of f. The Lefschetz sequence of f is defined to be: Lf ,Lf2,L f3,&ldots; A. Dold has proved that if the fixed point set of fn is compact for all n and s : N → Z is the Lefschetz sequence of f, then n| Ms(n) for all n. In this thesis, we investigate the number theoretic consequences of sequences which satisfy this property (called Dold sequences). In particular, we will investigate periodic Dold sequences. The main theorem states that a Dold sequence is periodic with period m if and only if Ms (n) = 0 for almost all n ∈ N and m = lcm{k ∈ N : Ms(k) ≠ 0}. Moreover, it is shown that a Dold sequence is bounded if and only if it is periodic. This extends a result of Babenko and Bogatyi˘.;The analysis of periodic Dold sequences is then applied to the study of mapping class groups of surfaces. Fuchsian groups are used to find all periodic Dold sequences of periodic orientation preserving maps on a surface. The solution of this realization problem provides some insight into the defect of the surjection Mod(S) → Sp(2 g, Z ) from preserving Nielsen-Thurston type.;Finally, algebraic number theory is used to determine a necessary and sufficient condition that a Zp -action (p an odd prime) on a closed connected oriented surface extends to the handlebody which it bounds. This analysis results from investigating the Atiyah-Singer g-Signature Theorem. The main theorem states that the equivariant signature is 0 if and only if action extends. |
