What happened to Jesus's cross after he died?

Based on what I have read, the cross was left at the site where the crucsifiction happened and was rediscovered in 326 by Constantine's mother Helena. As said, only a small fragment of that ancient wood remains and is highly guarded and kept under wraps in a church in Rome, possibly the Vatican. I took a visit to the Dom Cathedral in Cologne, Germany. We went inside the Treasury, and were stunned to find that behind a glass box, surrounded in pure gold and silver as well as precious jewels, there were actual fragmented remains of the Holy Cross. The subtext on the wall said that it was a gift from Emperor Alexis I of Constantinople. It was altared in the 13th century. (The cathedral is the largest in all of Germany, also being the 2nd tallest in Europe. If you havent seen it in person, scribble it on your things-to-do-before-you-die list. It's breathtaking.)