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Andrius Kulikauskas
 m a t h 4 w i s d o m  g m a i l
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 My work is in the Public Domain for all to share freely.
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 {$S^n\wedge S=S^{n+1}$} how does this relate to the two different ways of growing spheres?
 In topology product rule d(MxN) = dM x N union MxN addition is union (whereas in the Zariski topology multiplication is union). Why? The product rule is related to the deRham cohomology.
 What happens to the corners of the shapes?
 What is the topological quotient for an equilateral triangle or a simplex?
 Topological product (for a torus) is a list, has an order. In general, a Cartesian product is a list. What if such a product is unordered? How do we get there in the limit to F1?
 How can you cut in half a topological object if you have no metric? How can you be sure whether you will get two or three pieces?
 Try to imagine what a 3sphere looks like as we pass through it from time t = 1 to 1.
 What if there is a handle (a torus) inside a sphere? How to classify that?
 How is homotopy type theory related to dependent type theory and algebraic topology?
Videos
Books
 Allen Hatcher. Algebraic Topology  free on his website
 TaiDanae Bradley, Bryson, Terilla. Topology: A Categorical Approach.
 TaiDanae Bradley, Bryson, Terilla. Chapter 1. Examples and Constructions.
 TaiDanae Bradley, Bryson, Terilla. Chapter 4. Categorical Limits and Colimits.
 TaiDanae Bradley, Bryson, Terilla. Chapter 5. Adjunctions and the CompactOpen Topology.
 TaiDanae 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
Wikipedia
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.
ChernSimons theory
Hatcher exercise
Lebesgue covering dimension A way of defining dimension.
Unclear whether the empty space is path connected.
Point set topology
Division algebras
