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Homotopy Equivalences

by ibobev on 6/20/25, 9:56 AM with 24 comments

• by skulk on 6/23/25, 1:15 PM

  > In fact such a 2-sphere can be wrapped around the core an arbitrary number of times.

  This is really hard for me to visualize. What does it look like for a 2-sphere to wrap around the core multiple times? Also, I would have expected it to be able to wrap around in multiple ways since there are more dimensions here, leading to pi^2(b^3 \ {0}) = Z^2. How would one even prove that this isn't the case?

• by hackandthink on 6/23/25, 5:30 PM

  "Grothendieck conjectured that the infinity groupoid captures all information about a topological space up to weak homotopy equivalence"

  The homotopy hypothesis has something mystical about it.

  https://math.ucr.edu/home/baez/homotopy/homotopy.pdf

• by coderatlarge on 6/23/25, 1:20 PM

  in terry tao’s recent interview with lex fridman there’s an interesting bit on poincaré conjecture where he goes out of his way not to use these words.

• by m_j_g on 6/23/25, 6:47 PM

  vaguely related : synthetic homotopies visualisation tool - https://github.com/marcinjangrzybowski/cubeViz2