| Euclidean spaces,spheres,real projective spaces,complex projective spaces are very well-known spaces in the field of geometry and topology.They have very good topological and geometric properties.They are not only topological man-ifolds,but also differentiable manifolds,i.e.,topological manifolds with smooth structures.Many books use these four kinds of spaces as examples when intro-ducing a topological or geometric property.However,we cannot find one reference with a complete account of the important topological and geometric properties of these four kinds of spaces.Some properties were mentioned somewhere with no proof.Based on the importance of these spaces,this thesis gives a comprehensive and complete overview and proof of the topological and geometric properties of these spaces.These important topological properties include:their fundamental groups and the universal covering spaces,their homology groups and cohomology groups with Z and R coefficients,respectively.In the area of geometry,people s-tudy manifolds with some kind of structures.Manifolds with symplectic structures are called symplectic manifolds.Symplectic geometry originated from classical me-chanics and optics.Not all of Rn,Sn,RPn,CPn have symplectic structures.Using symplectic geometric method,this thesis discusses and proves which of them have symplectic structures.The content consists of the following parts?.The definitions of fundamental groups,covering spaces and their properties The fundamental groups and universal covering spaces of Rn,Sn,RPn,and CPn(n?1).?.The definition of the homology group of topological spaces and some ax-ioms.The homology groups with Z and R coefficients of Rn,Sn,RPn,and CPn(n?1).?.The definition of cohomology groups of topological spaces.The cohomol-ogy groups with Z and R coefficients of Rn,Sn,RPn,and CPn(n? 1).?.The definition and necessary conditions of symplectic manifolds.The discussion of whether Rn,Sn,RPn,and CPn(n? 1)have symplectic structures. |
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