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Allow me to provide a better brain teaser. Try to spot the mistake in this argument: If we delete an open disk from the 2Torus, then its fundamental group will be isomorphic to , where a and b are loops about the longitudinal and meridial circles and c is a loop about the boundary of the deleted disk... (More coming)

@Euler, but also note that our space deformation retracts onto the 1skeleton of the torus, and thus the homomorphism induced by the inclusion of that space is an isomorphism, so that is isomorphic to . Yet, this implies that c must homotopic to a constant loop, which means that is ZxZ. Yet, is the free group on 2generators, which is not isomorphic to ZxZ. So somewhere, we have a contradiction or this argument is invalid (it is invalid, try to spot the error).

This equation only works because it allows us to know that in this particular situation a=b but that's not always the case in other cases a=b=c=f so the equation fvcks off in all kinds of directions. Using variables in math to confuse people is the easiest thing to do, try proving your point with arc lengths and then you can talk shjt.
This is very false