دانلود کتاب Advanced Algebraic Topology

فهرست مطالب :

Introduction A recollection of category theory Categories Examples of categories Functors and natural transformations Universal properties and their applications A convenient category of topological spaces Top is not cartesian closed Compactly generated weakly Hausdorff spaces Homotopy groups Homotopy Group objects and co-group objects Algebraic structures on homotopy groups The fundamental groupoid Exact sequences of spaces Exact and co-exact sequences of topological spaces Mapping cones and the extended cofiber sequence Path spaces and the extended fiber sequence Relative homotopy groups n-connected maps Cofibrations and fibrations Hurewicz cofibrations Hurewicz fibrations Cofibrations and fibrations Serre fibrations and cofibrations CW-complexes CW-complexes Whitehead's theorem Simplicial and cellular approximation CW-approximation and homotopy excision CW-approximation of spaces The homotopy excision theorem The proof of homotopy excision Singular homology and cohomology The singular simplicial set of a space Singular (co)homology Brown representability and spectra Generalized homology and cohomology theories Examples Brown representability The stable homotopy category The stable homotopy category The smash product Constructing the stable homotopy category The Atiyah-Hirzebruch spectral sequence Cellular homology for generalized homology theories The Atiyah-Hirzebruch spectral sequence via exact couples Applications of the Atiyah-Hirzebruch spectral sequence The Atiyah-Hirzebruch-Serre spectral sequence The Atiyah-Hirzebruch-Serre spectral sequence First examples The cohomological Atiyah-Hirzebruch-Serre spectral sequence The cohomological Atiyah-Hirzebruch-Serre spectral sequence More examples Principal bundles and classifying spaces Principal G-bundles Transition functions and the bar construction Classifying spaces continued A characterization of universal bundles The bar spectral sequence Associated bundles and vector bundles Characteristic classes of vector bundles Characteristic classes Gysin sequences and Thom isomorphisms Chern classes and Stiefel-Whitney classes Bordism groups Bordism groups Bordism as a generalized homology theory The Pontryagin-Thom theorem The Pontryagin-Thom map The Pontryagin-Thom theorem Other tangential structures Steenrod operations Cohomology operations The Steenrod operations Constructing Steenrod operations First properties The Steenrod algebra The Adem relation The Steenrod algebra The cohomology of K(F2,n) Thom's theorem The cohomology of BO revisited The dual Steenrod algebra Thom's theorem Quasifibrations Thick geometric realization Quasifibrations Bott periodicity and topological K-theory Homology fibrations Harris' proof of Bott periodicity Topological K-theory The homotopy type of the cobordism category The cobordism category The Galatius-Madsen-Tillmann-Weiss theorem The K-theory of finite sets The scanning map Spaces of submanifolds The scanning map Outlook