These used to be on Alexander Kupers's webpage (http://people.math.harvard.edu/~kupers/teaching/231br/index.html), but they were removed. It's stated there that these are to be merged with those of the 2019 version of the course, so you may want to check there if there are newer versions first!
Author(s): Alexander Kupers
Year: 2018
Language: English
Pages: 314
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
