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Videos

• Pierre Albin. Algebraic Topology
• Richard Borcherds. Algebraic Topology
• Algebraic topology. July 2016. University of Chicago.
• Tadashi Tokieda, Topology and Geometry

Books

• Allen Hatcher. Algebraic Topology - free on his website
• Tai-Danae Bradley, Bryson, Terilla. Topology: A Categorical Approach.
  • Tai-Danae Bradley, Bryson, Terilla. Chapter 1. Examples and Constructions.
  • Tai-Danae Bradley, Bryson, Terilla. Chapter 4. Categorical Limits and Colimits.
  • Tai-Danae Bradley, Bryson, Terilla. Chapter 5. Adjunctions and the Compact-Open Topology.
  • Tai-Danae Bradley, Bryson, Terilla. Chapter 6. Paths, Loops, Cylinders, Suspensions...
• Peter May. A Concise Course in Algebraic Topology. Bott Periodicity in complex case.
• Peter May. More Concise Algebraic Topology: Localization, Completion, and Model Categories Bott periodicity in real case.
• Peter May. Simplicial Objects in Algebraic Topology. Discrete topology. Adjoint functors.
• Peter May. The Geometry of Iterated Loop Spaces.
• Ronald Brown. Topology and Groupoids: A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid. Chapter 9 gives an account of the Van Kampen theorem for adjunction spaces, computing the fundamental groupoid.
• Combinatorial Algebraic Topology
• Understanding Euler Characteristic, Ong Yen Chin
• Bott & Tu. Differential Forms in Algebraic Topology.
• William S. Massey. A Basic Course in Algebraic Topology. 1991.
• Tobias Berner. Introduction to Algebraic Topology
• William S. Massey, Algebraic Topology: An introduction (I think this book could be a helpful one)
• Ronald Brown. Topology and Groupoids. A geometric account of general topology. The utility of the algebra of groupoids for modeling geometry. The groupoid perspective is best for Van Kampen situations. Ronald Brown.

Articles

• Differential Forms in Algebraic Topology, Bott & Tu

Wikipedia

• The Geometrization conjecture and the eight Thurston geometries. Also, the Bianchi classification of low dimensional Lie algebras.

Ideas

• Tadashi Tokieda: Basic strategy of topology. When a problem has degeneracies, then deform (or perturb) to a problem without degeneracies, then deform back. We can use the same approach to show some problems are unsolvable.
• Quotient is gluing is equivalence on a boundary. Topology is the creation of a smaller space from a larger space.
• If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
• Constructiveness - closed sets any intersections and finite unions are open sets constructive
• A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other.
• Topology: invariants under smooth deformation. Thus, in a sense, different causes have the same effect, a form of symmetry but on a different level.
• Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole.
• Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1.
• Topology - getting global invariants (which can be calculated) from local information.
• Fundamental group gives useful information about a space. Classifying compact surfaces.
• In linear algebra, the dimension n of the space is irrelevant. But in algebraic topology the dimension is essential.

Chern-Simons theory

• Dijkgraaf-Witten theory is to be thought of as the finite group version of Chern-Simons theory.
• In topology, is the asymmetry between arbitrary unions and finite intersections the foundation for the difference between local and global?
• https://math.stackexchange.com/questions/69698/wedge-sum-of-circles-and-the-hawaiian-earring
• Armstrong. Basic topology.
• Munkres. Topology.
• James Munkres. Elements of algebraic topology.
• Sidney Morris. Topology Without Tears
• Brown. Topology and Groupoids.
• Resolution of singularities (in algebraic geometry) relates to universal covering of covering space (in algebraic topology).
• Semilocally simply connected -> can have a simply connected covering space - won't run into Zeno's paradox, which converts the diminishing sizes into the same size (of the names)
• https://en.wikipedia.org/wiki/Dunce_hat_(topology)
• https://en.wikipedia.org/wiki/Analytic_torsion
• How are algebraic topology and geometry related? Algebraic topology describes what is not there - the holes.
• Shmuel Weinberger. Algebraic theory of topology. Manifolds and K-theory: the legacy of Andrew Ranicki
• Think of a universal covering space as expressing the unfolding of a space, thus expressing eternal life.
• Covering spaces with repetition yield the spaces they cover.

Hatcher exercise

• https://math.stackexchange.com/questions/1427953/regarding-an-exercise-of-hatcher-fundamental-group-of-mathbbr3-with-n-l
• https://pages.uoregon.edu/njp/hw3solutions.pdf
• https://math.stackexchange.com/questions/939856/every-group-is-a-fundamental-group

Lebesgue covering dimension A way of defining dimension.

Unclear whether the empty space is path connected.

• Are the conditions for coverings the basis for completeness?
• The Fields Institute. Workshop on the Applications of Topology to Quantum Theory and Behavioral Economics.

Point set topology

• Heine-Borel theorem Proof of Heine-Borel theorem.
• Manya Raman-Sundstrom. A pedagogical history of compactness

Division algebras

• Allen Hatcher. Algebraic topology][[http://pi.math.cornell.edu/~hatcher/AT/ATpage.html | Explanation Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finite-dimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity.