## O Teorema do h-cobordismo e a conjectura generalizada de PoincarÃ

2005

##### RESUMO

The work of the following way In the first chapter it is developed to a list definitions and theorems that we found but important on subjects such as Topology, Differential Manifolds and aspects of the Algebraic Topology, which will be used in the following chapters. We considered advisable to indicate the demonstrations of the theorems, thus, like commentaries of the theory in the mentioned corresponding bibliografy ofter each result. The second chapter is dedicated to study the structure cobordism. In section 2.1 we introduce the cobordism concept and we make a summary of the general properties of functions of Morse on cobordism. With the definitions and results of section 2.1, we constructed in 2.2 section a topology that will allow to show the existence us of functions of Morse on a cobordism and will define this way the number of Morse of a cobordism. Functions of Morse provide as well in 2.3 vectorial fields of type gradient, fundamental result that it will help us to prove the theorem of the neighborhood necklace, that as well, will allow us to introduce an operation between cobordism. In section 2.3 also we will demonstrate that a cobordism with number of Morse zero, is indeed a cobordism product (theorem 2.3.3). This result is excellent in the theorem of the h-cobordism. The section 2.4 is dedicated to the study of cobordism with number of Morse 1. The homology of such cobordism will provide information on the decomposition with a cobordism in elementary cobordism. Finally, we finished the chapter 2 showing that any cobordism can be ordered like a composition of cobordism, where each cobordism has a function of Morse and a critical level, where each cobordism has a function of Morse and a critical level, where all their tactically important points have fixed index. This last result will be fundamental for the elimination of intermediary tactically important points of index (theorem 3.3.7). The third chapter is based on four fundamentals theorems that we can divide in two groups, the theorems 3.1.6, 3.2.4 and the theorems 3.3.7 and 3.4.1. The weak of cancelation (theorem 3.1.6) it says to us that a cobordism, is a cobordism product when we can bind the tactically important points by an only trajectory. The problem occurs when the tactically important points are not bound necessarily by an only trajectory, the theorem 3.2.4 provides conditions of transversallity. In this sense, we will study the behavior of our spheres in an intermediary level between both critical levels by means of conditions of transversallity both spheres, of such form that we pruned to apply theorems 3.1.6. The theorem 3.3.7 gives conditions us of elimination of tactically important points of intermediary index, whose demonstration uses theorem 3.1.6 strongly. The theorem 3.2.4 will be fundamental in the demonstration of theorem 3.3.7. This as well, next to theorem 3.4.1 it will help to demonstrate the theorem of the h-cobordism. Finally, the chapter 4 is dedicated to our primary target, the theorem of the h-cobordism. We make a small change of the demonstration given in the [12](to see [10]) and like consequence the theorem of Smale and characterization of n-discs with n >5. In the demonstration of the theorem of the h-cobordism we will both use last fundamental theorems of chapter 3