If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

Made available in DSpace on 2016-06-02T20:28:32Z (GMT). No. of bitstreams: 1 DissESP.pdf: 548814 bytes, checksum: 8cbb8820d445001b090bfd696177d474 (MD5) Previous issue date: 2005-08-31 / Universidade Federal de Sao Carlos / The purpose of this work is to present a generalized version of the classic Ljusternik-Schnirelmann theorem given by H. Steinlein [16] by using the con- cept of genus (c.f. [20]) of a Hausdor® space M with a mapping f : M ¡! M which generates a free Zp-action over M and to estimate the value of the genus of Lk;p, a special space used to construct an upper bound for the genus of (M; f). / O objetivo desse trabalho é apresentar uma versão generalizada do teorema clássico de Ljusternik-Schnirelmann devida a H. Steinlein [16], usando o con-ceito de genus (c.f. [20]) de um espaço de Hausdor® M com uma função f : M ¡! M que gera uma Zp-espaço livre em M, bem como estimar o valor do genus de Lk;p, um espaço especial usado para majorar o genus de (M; f).

The three geodesics theorem of Lusternik and Schnirelmann asserts that for every Riemannian metric on the 2-sphere, there exist at least three embedded closed geodesics. In the process of determining the geodesics as critical points of the energy or length functional, a suitable method of curve shortening is needed. It has been suggested to use the so-called curve shortening flow as it continuously deforms smooth embedded curves while naturally preserving their embeddedness. In the 1980s, the investigation of the curve shortening flow began and a proof of the Lusternik-Schnirelmann theorem using the flow was sketched. We build upon these results. After introducing the curve shortening flow, we prove the well-known result that the geodesic curvature of a smooth embedded closed curve on a smooth closed two-dimensional Riemannian manifold decreases smoothly to zero, provided the curve evolves forever under the flow. From this, we prove subconvergence to an embedded closed geodesic, using mainly local arguments. After introducing, in the form of Lusternik-Schnirelmann theory, the topological machinery employed in the process of determining critical points of certain functions, we turn to the three geodesics theorem which we prove under a few assumptions. For the round metric on the 2-sphere, we deformation retract a suitable space of unparametrized curves onto a simpler space of which we determine the homology groups relative to a subspace which deformation retracts onto the subspace of point curves. As this yields three subordinate homology classes, proving the validity of Lusternik-Schnirelmann theory for the curve shortening flow and the length functional on our space of curves completes the proof.