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@jtbrazas | |||||
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Meet the Hawaiian mapping torus - the mapping torus of the shift map. First singular homology is infinite cyclic generated by inner loop. Can you find the non-trivial elements of H_2? If you look at the image, it might feel H_2 is trivial since there is no "enclosed space!" pic.twitter.com/Vk39my2UrW
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Akiva Weinberger
@akivaw
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22. pro |
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Let L be a loop in the Hawaiian earring and let S be the shift map. If S(L) is homologous to L, then we can construct an element of H_2 by sort of making a cylinder of L on one side and S(L) on the other, and join them at the edge because they're homologous, if that makes sense
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Akiva Weinberger
@akivaw
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22. pro |
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So now we need an L. If a_n is the loop around the nth circle, I think
a_1+a_2−a_1+a_3−a_2+a_4−a_3+…
works, but I'm having a hard time proving that it's not nullhomologous.
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