One place to find examples of inventions and drivers that allowed European expansion to happen is in Chapter 3. There, Diamond describes various factors that allowed the Europeans to explore and to conquer those whom they found.
One of the major inventions that allowed Europeans to expand was steel. The Europeans had steel swords and steel armor. These things made them nearly invincible to many of the people they encountered when they were exploring.
A second invention that allowed them to explore was what Diamond refers to on page 78 as “maritime technology.” One type of maritime technology is, of course, ships. The Europeans had large ships that could sail across oceans while many of the people whom they encountered had nothing more than relatively small canoes. This allowed the Europeans to go out exploring across open oceans.
Finally, the invention of writing actually helped the Europeans to explore. It did so in a number of ways. For one thing, it helped allow the Europeans to have strong political organizations. The political organization allowed the Europeans to do a good job of gathering the money that was needed to make the ships and hire the people to sail them. The writing also allowed the Europeans to leave accounts of what they had done in the past. These included things like sailing instructions from past voyages. It also included accounts of what the explorers had seen. The ability to write, then, allowed the Europeans to tell each other how to get to various places and why they might want to go there. This would then allow the Europeans to continue to explore past the places they had already been told about.
Thus, we can see that steel, ships, and writing were all inventions that drove European exploration.