What technology/knowledge could you bring to the Middle Ages?

[TheRocketSurgeon]

They are not ready to use hygiene, much less steam engine, if these were dropped before their faces.  The Romans had both hygiene and labor saving technologies like large scale waterworks.   Men besotted by high Christianity had no use for them.

If I can bring a technology back, it would have to be a disruptive technology that they would actually strongly be tempted to use.    As with any fellowers of religions of piece, men of the middleages would enjoying destroying the works of their fellowmen second only to killing their fellow men.

So I would bring back gunpowder.   This I can guaranty they would use, and it would let the peasants topple men on horses and hasten the end of the disgusting middleages.

They had gunpowder in the mid-to-late Middle Ages, after trade contact with the far East and through the Muslim nations. In fact, at the seige of Harfleur, depicted in Shakespeare's Henry V, a cannon made the famous breach in the wall that the king exhorted his men to enter: "once more into the breach... or let us close up the wall with our English dead." And even if you brought back knowledge of how to make guncotton or some other more-modern powder, they simply lacked the metallurgical technology through most of the MA to allow anyone to make the necessarily-precise castings to use the modern powders in modern bullets (or to make new barrels/receivers/springs, if you melted yours or otherwise damaged the weapon I'm assuming you'd bring back with you).

Odds are, if you brought the Scientific Method back, you'd have to be particularly careful about to what year it was taken. The "dark ages" did not occur from a lack of knowledge, which is why the old "Roman engineering" ruins became, well, ruins. If you seemed to know too much, you would stand a good chance of making a religious leader feel threatened with your knowledge, and would wind up tied to a stake above a very warm patch of kindling. Or just stoned to death. And not in the good way.

I'd bring back solid knowledge of the printing press, about 1000 years before Gutenberg (1440 C.E.), and do my best to encourage literacy during the declining years of the Roman Empire but before the full rise of the priesthood's total dominance of culture. I would be careful to work with the church to print the Vulgate en masse, and if I could not receive permission from the Pope to use "my" invention on Latin Vulgate Bibles on behalf of the church, I would try to print things like Farmer's Almanacs which contained tidbits of scientific thinking that would be unlikely to land me in hot water--or hot kindling--with authorities.

Edit to Add: I see from the OP that I don't get a choice of year... it's the 11th century or nothing. Got it. Still would bring back the Press, and would be even more careful about what I wrote.