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Cantor Set of Algebraic Topology
I think that the Hawaiian Earring is the Cantor set of Algebraic Topology---key to lot of crazy examples. Fundamental group is not the free product of circles, not locally simply connected, the cone on it is not semilocally simply connected, the cone on it does not have a Universal Cover, you name it.
TLDR: Hawaiian Earring rocks!
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On Automorphism of covering spaces
Just had a marvelous aha-moment while doing Algebraic Topology. On Automorphism group where non-identity elements do not have a fixed point knowing a where a point get mapped shall completely determine the automorphism.
(Used to wonder, where this fixed point business is going.)
An interesting experiment conducted on a Mobius Strip.
Classical differential geometers (and classical analysts) did not hesitate to talk about "infinitely small" changes dx^i of the coordinates x^i, just as Leibnitz had. No one wanted to admit that this was nonsense, because true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the right way.)
Micheal Spivak, A comprehensive introduction to Differential Geometry, Vol 1.
Here is one of my favorite spaces: The Hawaiian earring, the "shrinking wedge of circles." This space is the first step into the world of "wild" topological spaces. This post is meant to be an intr...
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Slick proof that the commutator subgroup is normal.
Examples of Torsion free free-abelian groups.
A cult word of sixties, "grok" was coined, puportedly as a word from the Martian language, by Robert A. Heinlein in his pop science fiction novel 'Stranger in a Strange Land'. Its sense is nicely conveyed by the definition in 'The American Heritage Dictionary': "To understand profoundly through intuition and empathy".
Micheal Spivak, A Comprehensive Introduction to Differential Geometry, Vol 1.
“... you should not read further until you grok it.”
Cool demonstration.
We will be reading John Lee "Introduction to Manifolds" starting Dec. 21,2015.
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Alexander Horned Sphere
If our universe were nonorientable, then an astronaut who made a journey along some orientation-reversing path would return to earth with the right and left sides of his body interchanged: His heart would now be on the right side of his chest, etc.
William S. Massey, A Basic Course in Algebraic Topology.
Mobius loop, analytically.
Bertrand Russel on religion.
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A supercool demonstration and some insights on turning a Sphere inside out.
Isometric deformation of a Catenoid to Helicoid. There is an example in Do Carmo’s Differential Geometry book that illustrates that these two surfaces can be parametrized in such a way so that their first fundamental forms are equal (and hence they are locally isometric.)