Advanced Algebraic Topology

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