Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CONCERNING INVERSIBLE FIBRATIONS, GLOBALIZATION AND TRIVIALITY OF MAPS NIGAR TUNCER OZARSLAN SAMUEL EILENBERG CENTENARY CONFERENCE,2013 WARSAW, POLAND SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND NIGAR T. OZARSLAN Question Let be a locally trivial mapping and suppose that B is contractible. Does it follow that p is trivial? Definition The mapping is called trivial if there exists a homeomorphism for some space such that Corollary Assume that the space is paracompact, Hausdorff and locally contractible in the large. Then is an inversible fibration if and only if is locally trivial. E. Dyer and S.Eilenberg, Globalizing fibrations with schedules, Fund. Math. 130 (1988) SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND NIGAR T. OZARSLAN Definition Let be a map and be a space over and . Let such that . is called Hurewicz fibration if and only if there exists a map such that is the identity map Definition Let denote the set of all real numbers The path space of a topological space is the subspace of defined by Let be a space over . The space is the subspace of of all pairs such that A mapping is a D&E fibration if and only if there exists a continuous function (called action) To be written which satisfies the conditions and , where SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND NIGAR T. OZARSLAN A mapping is called inversible fibration if and only if there exists two reciprocal actions satisfying for all defined as Where the inverse path is SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND NIGAR T. OZARSLAN Regularity, inversibility, D&E fibrations, Hurewicz fibration Definition A lifting function is called regular if it lifts constant paths to constant paths. Definition An action is called regular action if and only if it satisfies the condition for all where Definition A map is an inversible Hurewicz fibration if and only if there exists two lifting functions satisfying . for every Definition A space over B is called regularly inversible Hurewicz fibration if and only if it is an inversible Hurewicz fibration with reciprocal regular lifting functions. SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND NIGAR T. OZARSLAN Theorem A space over B is an inversible D&E fibration then it is inversible Hurewicz fibration. Theorem A space over B is a regular D&E fibration if and only if it is a regular Hurewicz fibration. Theorem A space is regularly inversible D&E fibration if and only if it is regularly inversible fibration. Theorem If B is paracompact and contractible and f is a space over B then the followings are equivalent; (1) F is locally trivial, (2) F is an inversible D&E fibration, (3) F is an inversible Hurewicz fibration, (4) f is trivial. SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND THANK YOU NIGAR T. OZARSLAN