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)?”

In his thesis we are concerned with certain numerical invariants of homotopy type akin to the Lusternik-Schnirelmann category and cocategory. In a series of papers I. Bernstein, T. Ganea, and P.J. Hilton developed the concepts of the category and weak category of a topological space. They also considered the related concepts of conilpotency and cup product length of a space and the weak category of a map. Later T. Ganea gave another definition of category and weak category (which we shall write as G-cat and G-wcat) in terms of vibrations and cofibrations and hence this dualizes easily in the sense of Eckmann-Hilton. We find the relationships between these invariants and then find various examples of spaces which show that the invariants are all different except cat and G-cat. The results are contained in the following theorem. The map $e:B -> OmegaSigma B$ is the natural embedding. All the invariants are normalized so as to take the value 0 on contractible spaces. THEOREM Let B have the homotopy type of a simply connected CW-complex, then $cat B = G-cat B geq G-wcat B geq wcat B geq wcat e geq conil B geq cup-long B$ and furthermore all the inequalities can occur. All the examples are spaces of the form $B = S^qcup_alpha e^n$ where $alphain pi_{n-1} (S^q)$. When B is of this form, we obtain conditions for the category and the weak categories of B to be less than or equal to one of the terms of Hopf invariants of $alpha$. We use these conditions to prove the examples. We then prove the dual theorem concerning the relationships between the invariants cocategory, weak cocategory, nilpotency and Whitehead product length. THEOREM Let A be countable CW-complex, then $cocat A geq wcocat A geq nil A geq W-long A$ and furthermore all the inequalities can occur. The proof is not dual to the first theorem, though the examples we use to show that the inequalities can exist are all spaces with two non-zero homotopy groups. The most interesting of these examples is the space A with 2 non-zero homotopy groups, $mathbb Z$ in dimension 2 and ${mathbb Z}_4$ in dimension 7 with k-invariant $u^4 in H^8(mathbb Z, 2; {mathbb Z}_4)$. This space is not an H-space, but has weak cocategory 1. The condition $wcocat A leq 1$ is equivalent to the fact that d is homotopic to 0 in the fibration $D -d-> A -e-> OmegaSigma A$. In order to show that wcocat A = 1 we have to calculate to cohomology ring of $OmegaSigma K(mathbb Z,2)$. The method we use to do this is the same as that used to calculate the cohomology ring of $OmegaSigma S^{n+1}$ using James' reduced product construction. Finally we show that for the above space A the fibration $Omega A -g-> A^S -f-> A$ has a retraction $ ho$ such that $ hocirc g$ is homotopic to 1 even though A is not an H-space.

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)?”

Neste trabalho, mostramos a existência e multiplicidade de soluções para a seguinte classe de equações elípticas quasilineares { - \'DELTA IND. \'NÜ\' POT. \'upsilon\' + \'|\'upsilon\'| POT. \'NÜ\' - 2 \'upsilon\' = f(x, u), \'upsilon\' \'DIFERENTE\' 0, \'upsilon\' \'PERTENCE A >>: Nu + jujN2 u = f(x; u); x 2 ; u 6= 0; u 2 W1;N( ); onde e um domnio em RN, N 2, N e o operador N-Laplaciano e f e uma func~ao que possui um crescimento crtico exponencial. Para obter nossos resultados utilizamos o Princpio Variacional de Ekeland, Teorema do Passo da Montanha, Categoria de Lusternik- Schnirelman, Ac~ao de Grupo e tecnicas baseadas na Teoria do G^enero. Palavras chaves: Problemas elpticos quasilineares, Metodo Variacional, N-Laplaciano, crescimento crtico exponencial, Princpio Variacional de Ekeland, Categoria de Lusternik- Schnirelman, Desigualdade de Trudinger-Moser / In this work, we show the existence and multiplicity of solutions for the following class of quasilinear elliptic equations { - \'DELTA\' IND. \'NÜ\' \'upsilon\'\' + |\'upsilon\'| POT. \'NÜ\' - 2 = f(x, \'upsilon\'), x \"IT BELONGS\' \'OMEGA\', \'upsilon\' \'DIFFERENT\' 0, \'upsilon\' \'IT BELONGS\' W POT. 1, \'NÜ\' ( OMEGA), where \'OMEGA\' is a domain in \' R POT. \'NÜ\' > OR = 2, \'DELTA\' IND. \'NÜ\' is the N-Laplacian operator and f is a function with exponential critical growth. To obtain our results we utilize the Ekeland Variational Principle, the Mountain Pass Theorem, Lusternik-Schnirelman of Category, Group Action and techniques based on Genus Theory

Neste trabalho, mostramos a existência e multiplicidade de soluções para a seguinte classe de equações elípticas quasilineares { - \'DELTA IND. \'NÜ\' POT. \'upsilon\' + \'|\'upsilon\'| POT. \'NÜ\' - 2 \'upsilon\' = f(x, u), \'upsilon\' \'DIFERENTE\' 0, \'upsilon\' \'PERTENCE A >>: Nu + jujN2 u = f(x; u); x 2 ; u 6= 0; u 2 W1;N( ); onde e um domnio em RN, N 2, N e o operador N-Laplaciano e f e uma func~ao que possui um crescimento crtico exponencial. Para obter nossos resultados utilizamos o Princpio Variacional de Ekeland, Teorema do Passo da Montanha, Categoria de Lusternik- Schnirelman, Ac~ao de Grupo e tecnicas baseadas na Teoria do G^enero. Palavras chaves: Problemas elpticos quasilineares, Metodo Variacional, N-Laplaciano, crescimento crtico exponencial, Princpio Variacional de Ekeland, Categoria de Lusternik- Schnirelman, Desigualdade de Trudinger-Moser / In this work, we show the existence and multiplicity of solutions for the following class of quasilinear elliptic equations { - \'DELTA\' IND. \'NÜ\' \'upsilon\'\' + |\'upsilon\'| POT. \'NÜ\' - 2 = f(x, \'upsilon\'), x \"IT BELONGS\' \'OMEGA\', \'upsilon\' \'DIFFERENT\' 0, \'upsilon\' \'IT BELONGS\' W POT. 1, \'NÜ\' ( OMEGA), where \'OMEGA\' is a domain in \' R POT. \'NÜ\' > OR = 2, \'DELTA\' IND. \'NÜ\' is the N-Laplacian operator and f is a function with exponential critical growth. To obtain our results we utilize the Ekeland Variational Principle, the Mountain Pass Theorem, Lusternik-Schnirelman of Category, Group Action and techniques based on Genus Theory

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.