Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Introduction E9F5FC Questions FFFFC0 Software 代数拓扑 {$S^n\wedge S=S^{n+1}$} 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? Videos Books Allen Hatcher. Algebraic Topology - free on his website Tai-Danae Bradley, Bryson, Terilla. Topology: A Categorical Approach. 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 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.
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