04847nam 2201189 a 450 991081354120332120230803020246.01-4008-4652-81-299-05144-810.1515/9781400846528(CKB)2550000000996711(EBL)1114887(OCoLC)828077867(SSID)ssj0000822546(PQKBManifestationID)11442046(PQKBTitleCode)TC0000822546(PQKBWorkID)10760684(PQKB)10049062(MiAaPQ)EBC1114887(StDuBDS)EDZ0001756468(DE-B1597)453494(OCoLC)933516592(OCoLC)990531966(DE-B1597)9781400846528(Au-PeEL)EBL1114887(CaPaEBR)ebr10652016(CaONFJC)MIL436394(EXLCZ)99255000000099671120121220d2013 uy 0engurcn|||||||||txtccrSpaces of PL manifolds and categories of simple maps /Friedhelm Waldhausen, Bjørn Jahren and John RognesCourse BookPrinceton Princeton University Press20131 online resource (193 p.)Annals of Mathematics Studies ;210Annals of mathematics studies ;no. 186Description based upon print version of record.0-691-15775-8 0-691-15776-6 Includes bibliographical references and index.Front matter --Contents --Introduction --1. The stable parametrized h-cobordism theorem --2. On simple maps --3. The non-manifold part --4. The manifold part --Bibliography --Symbols --IndexSince its introduction by Friedhelm Waldhausen in the 1970's, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a "desingularization," improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.Annals of Mathematics StudiesPiecewise linear topologyMappings (Mathematics)CAT Whitehead space.CAT manifold.Euclidean neighborhood retract.Euclidean polyhedra.PL Serre fibration.PL Whitehead space.PL homeomorphism.PL manifold.Quillen's Theorem A.Quillen's Theorem B.absolute neighborhood retract.algebraic K-theory of spaces.algebraic K-theory.approximate lifting property.cofibration.degeneracy operator.desingularization.finite simplicial set.h-cobordism theorem.homotopy equivalence.homotopy fiber sequence.horn.manifold space.manifold.n-manifolds.normal subdivision.polyhedral realization.q-simplex.reduced mapping cylinder.simple map.simplicial set.space.stabilization map.stable parametrized h-cobordism theorem.tangent microbundle.thickening.triangulation.weak homotopy equivalence.Čech homotopy type.Piecewise linear topology.Mappings (Mathematics)514/.22SI 830rvkWaldhausen Friedhelm1938-1686918Jahren Bjørn1945-1686919Rognes John521348MiAaPQMiAaPQMiAaPQBOOK9910813541203321Spaces of PL manifolds and categories of simple maps4060010UNINA