What is incumbent upon a man who said to another, "She is unlawful to me from her back," meaning his wife, after that man made him swear an oath concerning something he did and denied?

This phrasing is not explicit in establishing ẓihār (a type of divorce-like pronouncement) or ṭalāq (divorce), and its ruling depends on your intention. If you intended ẓihār and ṭalāq, and you were lying, then they are considered to have occurred according to the majority of scholars. This is contrary to Sheikh al-Islam Ibn Taymiyyah, who holds that it is an oath requiring only repentance. However, if you were truthful or swore by what you believed to be correct, then nothing is binding upon you. One must be cautious when using such phrases.