I know that the the PIE language had words for 'city' and 'town'. With the PIE language, their vocabulary can be used to figure out things about what they had; words for 'wheel' likely meant they had wheels, words for crops likely meant they had an understanding of agriculture. Did the PIE people build their own cities, have knowledge of the cities in cultures elsewhere, or was it simply a coincidence that the root 'polis' came to mean 'city' all across their diaspora?
it simply a coincidence that the root 'polis' came to mean 'city' all across their diaspora?
The premise here is false. Per Buck's Dictionary of Selected Synonyms in the Principal Indo-European Languages, the root is reflected in Greek, yes, and in Sanskrit as pur 'stronghold, castle', and perhaps is reflected by Lithuanian pilis 'castle', but there are over a dozen different etyma with the meaning 'town' or 'village'. While we can't use this evidence to rule out that PIE speakers had cities (that's really more a question for archaeologists--can we identify cities in the Pontic and Caspian steppes that date to about 6000 years ago, /u/brigantus?) it's consistent with the hypothesis that no, they didn't: the large number of sources for words with this meaning suggests that they were independently invented, usually from words having something to do with enclosure (e.g. English town from Old English 'enclosure, garden', Proto-Slavic *gordъ from a separate PIE root meaning 'enclose, circle, gird'). It's a bit hard to count all the separate items given but I think you're looking at about 17 different sources for this semantic family of words.
Compare the plethora of etyma for 'city, town' with the 2 etyma we have for 'wheel' in basically every attested Indo-European language: *kʷekʷlo- (from the root *kʷel- 'turn end-over-end') and *hrot- 'roll'. These two etyma get us (by a process of regular and exceptionless sound change) Ancient Greek kyklos, Norse hvel and hjol, English wheel and Dutch wiel, Old Church Slavonic kolo and descendants, Sanskrit cakra-, Avestan čaxra, Latin rota and descendants in Romance, Irish roth, Welsh rhod, German rad, and Lithuanian ratas. Now, that tells you you're looking at something you can really securely reconstruct to PIE.