Four-Dimensional Poincar(?) Conjecture Proof And Its Influence On Mathematics And Physics

Although the 3-dimensional Poincar(?) conjecture has been the focus of academic attention for the past decade because of its difficulty,the length of time it takes to solve it,and its relevance to the shape of the universe.But to try to observe and imagine the overall shape of the universe,we should be in at least 4-dimensions;In addition,we live not only in a three-dimensional universe,but also in a 1-dimensional,or 4-space-time world,so the 4-dimensional Poincar(?) conjecture is the most relevant to understanding the shape of the universe and the world we live in.And so it is.The proof of the 4-dimensional Poincar(?) conjecture has advanced the research of 4-manifold and Gauge Theory.Therefore,it is necessary to analyze in detail the origin,proof and the whole process of establishing the relation with the 4-manifold and Gauge Theory of the 4-dimensional Poincar(?) conjecture,so as to promote the understanding of the mathematical and physical significance of the four-dimensional pungalay conjecture.Based on the original literature and the research literature,this paper discusses the two proofs of the 4-dimensional Poincar(?) conjecture and their influence on the 4-manifold and Gauge Theory.The thesis is divided into four chapters.The first chapter firstly examines the mathematical and physical background of the 4-dimension Poincar(?) conjecture through the historical background of the development of topology and Gauge Theory,especially the various proofs and inferences of other dimensions of Poincar(?) conjecture.Secondly,based on Freedman's growing environment and study experience,this paper discusses the reasons for Freedman's concern about the 4r-dimensional Poincar(?) conjecture.Finally,the work of Freedman to prove the 4-dimensional Poincar(?) conjecture using the technique of "Casson handle" is analyzed.The second chapter firstly explores the background of Donaldson's conjecture based on his growing environment and working experience.Secondly,it analyzes how he,as a mathematician,proves the 4-dimensional Poincar(?) conjecture again with the canonical Gauge Theory in physics.Finally,the author discusses the combination and supplement of his work with Freedman after this new method of proof.The third chapter first discusses the significance of the proof of the 4-dimensional Poincar(?) conjecture based on the research of Freedman and Donaldson.Secondly,based on this,this paper analyzes the advancement of Freedman and Donaldson's research on 4-manifold.The fourth chapter is based on the proof of the 4-dimensional Poincar(?) conjecture and the research of the 4-dimensional manifold,firstly,this paper explores Donaldson's research on Gauge Theory;secondly,it explores the advancement of the research of Topological Quantum Field Theory by Witten and Donaldson.