Homotopy group

Spotify is the famous viral media. Your MP3 file spread all over the world qickly because of the internet, although you need to be famous. We have different divices to listen our favorite music. This is addiction and virus actually. Their industry is huge, and Apple is popular.

This is vaccinated because of math. There is no rock star but logic.

You have a sphere which is the earth. You can use iPhone anywhere because of the universal network like Starlink.

Then you have a doughnut in Starbucks.

You can eat it, but you are not there. Starbucks isn't the doughnut. They are the different group. This is homotopy. The doughnut is called torus.

T is torus, and C is the complex plane.

∴

S is the earth. Therefore, this isn't real.

Moreover, This is metric.

w(f)∈Z

w(f)=w(g)

∥f∥=sup(｜f｜)

This is the maxium range.

∥f∥≦L and ∥g∥≦L

∥H∥≦L

H is homotopy.

However, when you have your CD, you need to buy it and have MP3. This is time which is the different dimension. You can't put your music on the web directly.

This is false.