Is stitching the shoulder considered part of the back, such that an oath sworn about my brother having surgery on his back would not be considered a lie, and what is the ruling on this oath, knowing that I was unaware of him having surgery on his back, and is this an oath (yamin ghamus) that necessitates expiation or not, and what are its consequences on the Day of Judgment?

The oath mentioned is a false oath. The majority of jurists hold that what is obligatory in this case is repentance and restoring rights to their owners. Al-Shafi'i and others are of the view that expiation is obligatory. It is best to combine repentance and expiation as a precaution.

As for the expiation for an oath, fasting is not sufficient unless one is unable to feed, clothe, or free a slave. There is no evidence linking the submerging oath (yameen ghamus) with deprivation from intercession.

And he who repents of a sin is like him who has no sin.