I'm assuming "third-wave era" is a technical term that's now in vogue for high-school world history courses in order to categorize stuff for world history, since it's clearly not [Alvin Toffler's](http://en.wikipedia.org/wiki/The_Third_Wave_(Toffler) Third Wave. The big problem with this kind of model is that it will probably ignore the Americas and Sub-Saharan Africa, which will not fit any type of progressive world history theories and models. You could call these groups anomalies or variations on the trend, but when 3 out of 5 continents don't meet the trend, is it really a trend?

So, if you are talking about European civilizations from 500-1500, this is the middle ages/medieval period. In China, it would be part of imperial China. And so on and so forth for other regions. These regions had their own religions, cultures, politics, and modes of thought -- all of which had their own goals and pursuits. Some of them interacted with each other, directly or indirectly.

When you say "catching up," it implies that one particular method or people is superior than the other. Such an evaluation can be made about things such as the tallest building or biggest temple, but would be endlessly contentious about anything involving historical analysis. I would also question what purpose such an inquiry would serve. The point of history is to investigate and understand the past, how does arguing which thing is best aid in that pursuit? It is dangerously close to a "what if?" scenario, which is not what historians do. Now, if the question was instead, "What cultures influenced medieval European civilization?" there are all sorts of interesting answers, most of which would be cross-cultural, not simply one group taking from another. This may seem semantic, but semantics are incredibly important in the study of history, since words from the past are often all we have to work with.

If you were educated in the U.S. or Europe, the focus on European civilizations is because of the direct cultural links these groups have to them. There is also an indisputable influence of western culture on modern global culture, so it is not surprising that people want to investigate how that happened. However, in the world of professional history, I would argue that history post-1500 (ignoring the problems of periodization) receives much, much more attention than 500-1500. Look at any university history department website, you'll likely find there are many more historians studying things outside of this time period. There used to be an idea of western exceptionalism, but this has been largely rejected by historians, though there are still competing ideas such as the Great Divergence. But again, these ideas occur largely outside of the scope of 500-1500.

Hope this helps a little. Happy reading!