Betti Numbers Of The Equal Value Surface Of A Bott Function

Bott functions is the generalization of Morse functions. Morse theory provides the relationship between the topological properties of a manifold and the critical points of a Bott function on the manifold.The Orbits of a Hamilton system lie on the equal energy surfaces of the system. So the topological properties of the orbits of a Hamilton system depend on the topological properties of the equal energy surface of the system. We can use Morse theory to study the topological properties of the equal energy surface of a Hamilton system, when the Hamilton function of the system is a Bott function. In the reference [13] Gu Zhi Ming used Morse inequality to get the estimate of the upper bound of the first Betti mumber of the equal energy surfaces of a Hamilton system.In the reference [4] Atiyah proved the connectivity of the equal value surfaces of the Boot functions whose indexes and coindexes are not equal to 1. The theorem played a key role in the proof of the convexity of a moment mapping.There is great importance in the study of Betti numbers of the equal value surfaces of the Bott functions. The paper is dedicated to the study of Betti numbers of the equal value surfaces of the Bott functions.The main results are as follows:1. I get a formula of relative homology groups with respect to the equal value surfaces of the Bott functions. Then I use the formula and the methods arising from the reference [13] to get the estimate of the upper bound and the lower bound of the Betti mumber of the equal value surfaces of a Bott function.2. I use Morse inequality and Lefschetz dualilty theorem to get a new proof of the connectivity of the equal value surfaces of the Boot functions whose indexes and coindexes are not equal to 1.And I give the proof of the connectivity of the equal value surface correspondence to its critical value.3. I use Mayer-Vietoris exact homology sequence and the interval principle of Morse functions to get an equality of the Betti numbers of the equal value surfaces of the Morse functions whose indexes are not odd.