What inventions led to the upward and outward spread of cities?

There were many. So I will list some for you:

Steel: First efficiently manufactured by Andrew Carnegie, this allowed stronger buildings to be made, which led to Skycrapers in many cities.

Railroad: Invented in 1831, railroads allowed materials to be transported much faster than traditional methods.

Elevators: If it weren't for this invention, we wouldn't have 100 story high skycrapers. There would be now way to get all the way up and down without taking the stairs, which can be extremely complicated and tiring.