Steel for spokes.

Rubber pneumatic tires.

Ball bearings.

Then if you want to get a safety bicycle rather than a penny farthing, you need a cheap process for bike chains.

Belts work for bike chains. Using those on steam engines.

Wikipedia seems to have modern steelmaking start in 1855 with the Bessemer process. I suspect that a decent quality of steel is needed for spokes, if nothing else. The safety bicycle seems like just a spontaneous invention, but was a huge boost to a already popular activity.

> I suspect that a decent quality of steel is needed for spokes,

That's a good guess, as "Wire Wheels" were first successfully used commercially for bicycles.

https://en.wikipedia.org/wiki/Wire_wheel#On_bicycles

Not much has really changed since then. All my bikes have spoked wheels. Works great, light, relatively inexpensive.

> That's a good guess, as "Wire Wheels" were first successfully used commercially for bicycles.

We did have 'wagon' style wheels[0] before that, though, with spokes in compression, rather than tension.

[0] http://www.oldbike.eu/museum/wp-content/uploads/2011/08/1891...

Edit: A picture!