Topology problem

Level 1

Is there a homeomorphism between the two surfaces?

False True

Icaro Buscarini by Icaro Buscarini , 26, Brazil

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2 solutions

貓頭鷹 女皇
Aug 18, 2019

The fundamental group of a sphere and a torus are different

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Luca Sansilvestri
Oct 23, 2019

Both surface are connected so you can find their fundamental group, and you have the implication that if the two topological spaces are homotopic they have the same fundamental group, but you also have the implication that homeomorfic->homotopic. So we have homeomorfic->homotopic->same fundamental group, now we know from Seifert Van Kampen that the two surface don’t have the same fundamental group we also know that they are not homotopic and also not homeomorfic.

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The torus has a hole, but the sphere does not. No matter how hard you morph or squish it (without glueing or cutting because it must be homeomorphic/continous), it is impossible to transform a sphere into a torus (and vice versa).

Tin Long - 10 months ago

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