Homology Methods Of Topology And Applications

Topology is of fundamental significance to continuous mathematics.In addition to affecting many branches of mathematics,the concepts and methods of topology are also applied to interdisciplinary research,such as molecular topology structure,topoisomerase and liquid crystal structure defect classification.In recent years,homology theory in topology,as an important tool combined with computing methods,has been widely used in big data analysis.Combined with computing methods,linear regression and neural networks in machine learning,this thesis discusses applications of homology methods of topology in several aspects.The main contents include the following four parts:In the first part,as an application of cohomology,we discuss the bordism classifi-cation of manifolds with an involution fixing the product of a class of even dimensional real projective space and a class of complex projective space.In two specific cases,we answer an important question proposed by Steenrod,an authoritative expert in topol-ogy,in 1962.Firstly,we construct a manifold satisfying the requirements and prove the existence of bordant manifolds with the involution.Secondly,by analyzing the al-gebraic topological properties of fixed point set and its normal bundle,with the help of Kosniowski-Stong formula,we skillfully construct symmetric polynomials to prove the non-existence of non-bordant manifolds with the involution,and finally give the bordism classification of manifolds with the involution.In the second part,topological methods are applied to the study of fullerene molecules and their isomers,and relationships between their structures and energies are analyzed.The results generalize and deepen those of other experts such as Prof.Guowei Wei from Michigan State University.By extracting 10 topological features of fullerene molecules based on the newly developed persistent homology theory,and selecting appropriate ac-tive functions and parameters,a new neural network model PHNN(PH-based neural network)is established.Then relationships between the structures and energies of 500isomer molecules from 10 different fullerene families are analyzed.The effect of the model is verified by Pearson correlation coefficient which is better than that reported in previous literature.In the third part,persistent homology theory is used to analyze the endohedral metallofullerene molecules Ni@C_nfor the first time,and a linear regression model based persistent-homology(PH-based Linear Regression,PHLR)is established to analyze the relationships between the structures and energies of the endohedral metallofullerene molecules.The method used is to consider the topological structure of the endohe-dral metallofullerene molecules,to extract their topological features based on topological homeomorphism invariants,which are expressed in form of the average of barcode length,and then to analyze the energies and stability of the endohedral metallofullerene molecules Ni@C_n.In order to show the effect of the model,compared with the experimental result-s,we obtain the Pearson correlation coefficient 0.9997,indicating that there is a strong correlation between the energies of Ni@C_nand the energies predicted by PHLR model.So this is an ideal result.In the fourth part,the PHLR(PH-based Linear Regression)model is used to analyze the relationship between the structures and energies of metal cluster molecules Co_n for the first time.By using the selected topological feature index,that is,the average of barcode length,the energies and stability of the metal cluster molecules are analyzed.In order to show the effect of the model,compared with the experimental results,we obtain the Pearson correlation coefficient 0.9948,which is also very ideal.