Spectral theory is the study of Mark Kac's famous question [K], "can one hear the shape of a drum?" That is, can we determine the geometrical or topological properties of a manifold by using its Laplace Spectrum? In recent years, the problem has been extended to include the study of Riemannian orbifolds within the same context. In this thesis, on the one hand, we answer Kac's question in the negative for orbifolds that are spherical space forms of dimension higher than eight. On the other hand, for the three-dimensional and four-dimensional cases, we answer Kac's question in the affirmative for orbifold lens spaces, which are spherical space forms with cyclic fundamental groups. We also show that the isotropy types and the topology of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. This is done in a joint work with E. Stanhope and D. Webb by using P. Berard's generalization of T. Sunada's theorem to obtain isospectral orbifolds. Finally, we construct a technique to get examples of orbifold lens spaces that are not isospectral, but have the same asymptotic expansion of the heat kernel. There are several examples of such pairs in the manifold setting, but to the author's knowledge, the examples developed in this thesis are among the first such examples in the orbifold setting.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21545号 / 理博第4452号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 大槻 知忠, 教授 向井 茂, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Neste trabalho, mostramos que fluxos de Reeb dinamicamente convexos em um espaço lenticular $L(p, 1)$, $p>1$, admite uma órbita periódica de Reeb especial $P$ que é o binding de uma decomposição em livro aberto racional, com páginas tipo-disco tal que cada página é uma seção global. O índice de Conley-Zehnder da $p$-ésima iterada de $P$ é $3$. Como corolário, o fluxo de Reeb possui duas ou infinitas órbitas periódicas. Este resultado aplica-se ao Hamiltoniano de Hénon-Heiles, cujo fluxo restrito a energia baixa possui $Z_3$-simetria e define um fluxo de Reeb em $L(3, 1)$. Devido a $Z_4$-simetria aplicamos nosso resultado ao problema lunar de Hill regularizado. Na segunda parte deste trabalho investigamos a existência de uma folheação $3-2^3$ em níveis de energia no sistema Hamiltoniano de Hénon-Heiles, para energia logo acima da crítica. Provamos que certa região de interesse é uma hipersuperfície de contato. Provamos também que o fluxo de Reeb possui uma órbita periódica $Z_3$ simétrica, cujo índice de Conley-Zehnder é $3$ e possui número de auto-enlaçamento $-1$. / We show that a dynamically convex Reeb flow on a lens space $L(p, 1)$, $p>1$ admits a special closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages so that each page is a global surface of section. The Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. As a corollary, the Reeb flow has $2$ or infinitely many closed Reeb orbits. This result applies to the Hénon-Heiles Hamiltonian whose flow restricted to low energy levels has $Z_3$-symmetry and descends to $L(3,1)$. Due to a $Z_4$-symmetry we also apply our results to Hill\'s lunar problem. In the second part of this work we investigate the existence of a $3-2^3$ foliation on energy levels of the Hénon-Heiles Hamiltonian, for energies above the critical one. We show that some region is of contact-type and the Reeb flow has a $Z_3$-symmetric periodic orbit, whose Conley-Zehnder is $3$ and has self-linking number $-1$.

Neste trabalho, mostramos que fluxos de Reeb dinamicamente convexos em um espaço lenticular $L(p, 1)$, $p>1$, admite uma órbita periódica de Reeb especial $P$ que é o binding de uma decomposição em livro aberto racional, com páginas tipo-disco tal que cada página é uma seção global. O índice de Conley-Zehnder da $p$-ésima iterada de $P$ é $3$. Como corolário, o fluxo de Reeb possui duas ou infinitas órbitas periódicas. Este resultado aplica-se ao Hamiltoniano de Hénon-Heiles, cujo fluxo restrito a energia baixa possui $Z_3$-simetria e define um fluxo de Reeb em $L(3, 1)$. Devido a $Z_4$-simetria aplicamos nosso resultado ao problema lunar de Hill regularizado. Na segunda parte deste trabalho investigamos a existência de uma folheação $3-2^3$ em níveis de energia no sistema Hamiltoniano de Hénon-Heiles, para energia logo acima da crítica. Provamos que certa região de interesse é uma hipersuperfície de contato. Provamos também que o fluxo de Reeb possui uma órbita periódica $Z_3$ simétrica, cujo índice de Conley-Zehnder é $3$ e possui número de auto-enlaçamento $-1$. / We show that a dynamically convex Reeb flow on a lens space $L(p, 1)$, $p>1$ admits a special closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages so that each page is a global surface of section. The Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. As a corollary, the Reeb flow has $2$ or infinitely many closed Reeb orbits. This result applies to the Hénon-Heiles Hamiltonian whose flow restricted to low energy levels has $Z_3$-symmetry and descends to $L(3,1)$. Due to a $Z_4$-symmetry we also apply our results to Hill\'s lunar problem. In the second part of this work we investigate the existence of a $3-2^3$ foliation on energy levels of the Hénon-Heiles Hamiltonian, for energies above the critical one. We show that some region is of contact-type and the Reeb flow has a $Z_3$-symmetric periodic orbit, whose Conley-Zehnder is $3$ and has self-linking number $-1$.

A simplicial cell complex K is the face poset of a regular CW complex W such that the boundary complex of each cell is isomorphic to the boundary complex of a simplex of same dimension. If a topological space X is homeomorphic to W then we say that K is a pseudotriangulation of X. For d 1, a (d + 1)-colored graph is a graph = (V; E) with a proper edge coloring : E ! f0; : : : ; dg. Such a graph is called contracted if (V; E n 1(i)) is connected for each color A contracted graph = (V; E) with an edge coloring : E ! f0; : : : ; dg determines a d-dimensional simplicial cell complex K( ) whose vertices have one to one correspondence with the colors 0; : : : ; d and the facets (d-cells) have one to one correspondence with the vertices in V . If K( ) is a pseudotriangulation of a manifold M then ( ; ) is called a crystallization of M. In [71], Pezzana proved that every connected closed PL manifold admits a crystallization. This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings. We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of vertices of any crystallization of a connected closed 3-manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3-manifolds RP3, L(3; 1), L(5; 2), S1 S1 S1, S2 S1, S2 S1 and S3=Q8, where Q8 is the quaternion group. Moreover, there is a unique such vertex minimal crystallization in each of these seven cases. We have also constructed crystallizations of L(kq 1; q) with 4(q + k 1) vertices for q 3, k 2 and L(kq +1; q) with 4(q + k) vertices for q 4, k 1. In [22], Casali and Cristofori found similar crystallizations of lens spaces. By a recent result of Swartz [76], our crystallizations of L(kq + 1; q) are vertex minimal when kq + 1 are even. In [47], Gagliardi found presentations of the fundamental group of a manifold M in terms of a crystallization of M. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M, we have constructed a crystallization of M. These results are in Chapter 3. We have de ned the weight of the pair (hS j Ri; R) for a given presentation hS j R of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of (hS j Ri; R) is n then our algorithm constructs all the n-vertex crystallizations which yield (hS j Ri; R). As an application, we have constructed some new crystallization of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For m 3 and m n k 2, our generalized algorithm gives a 2(2m + 2n + 2k 6 + n2 + k2)-vertex crystallization of the closed connected orientable 3-manifold Mhm; n; ki having fundamental group hx1; x2; x3 j xm1 = xn2 = xk3 = x1x2x3i. These crystallizations are minimal and unique with respect to the given presentations. If `n = 2' or `k 3 and m 4' then our crystallization of Mhm; n; ki is vertex-minimal for all the known cases. These results are in Chapter 4. We have constructed a minimal crystallization of the standard PL K3 surface. The corresponding simplicial cell complex has face vector (5; 10; 230; 335; 134). In combination with known results, this yields minimal crystallizations of all simply connected PL 4-manifolds of \standard" type, i.e., all connected sums of CP2, CP2, S2 S2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair 4-manifolds which are homeomorphic but not PL-homeomorphic. We have also presented an elementary proof of the uniqueness of the 8-vertex crystallization of CP2. These results are in Chapter 5. For any crystallization ( ; ) the number f1(K( )) of 1-simplices in K( ) is at least d+1 . It is easy to see that f1(K( )) = d+1 if and only if (V; 1(A)) is connected for each d 2 2 1)-set A called simple. All the crystallization in Chapter 5 (. Such a crystallization is are simple. Let ( ; ) be a crystallization of M, where = (V; E) and : E ! f0; : : : ; dg. We say that ( ; ) is semi-simple if (V; 1(A)) has m + 1 connected components for each (d 1)-set A, where m is the rank of the fundamental group of M. Let ( ; ) be a connected (d +1)-regular (d +1)-colored graph, where = (V; E) and : E ! f0; : : : ; dg. An embedding i : ,! S of into a closed surface S is called regular if there exists a cyclic permutation ("0; "1; : : : ; "d) (of the color set) such that the boundary of each face of i( ) is a bi-color cycle with colors "j; "j+1 for some j (addition is modulo d+1). Then the regular genus of ( ; ) is the least genus (resp., half of genus) of the orientable (resp., non-orientable) surface into which embeds regularly. The regular genus of a closed connected PL 4-manifold M is the minimum regular genus of its crystallizations. For a closed connected PL 4-manifold M, we have provided the following: (i) a lower bound for the regular genus of M and (ii) a lower bound of the number of vertices of any crystallization of M. We have proved that all PL 4-manifolds admitting semi-simple crystallizations, attain our bounds. We have also characterized the class of PL 4-manifolds which admit semi-simple crystallizations. These results are in Chapter 6.