Andrius Kulikauskas

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Introduction E9F5FC

Questions FFFFC0



  • {$S^n\wedge S=S^{n+1}$} 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 3-sphere 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?






  • 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

Hatcher exercise

Lebesgue covering dimension A way of defining dimension.

Unclear whether the empty space is path connected.

Point set topology

Division algebras

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This page was last changed on June 29, 2024, at 12:16 PM