Did Abraham write the Torah

It was Moses who wrote the complete Torah, shortly before he died (Deuteronomy 31).

Before Moses, they had Israelite parchments on which their earliest traditions were recorded (midrash Shemot Rabbah 5:18). These included the entire book of Genesis (ibid. 5:22), which had been prophetically written by the Forefathers themselves, including Abraham. The significant portions of the early parchments were incorporated into the Torah by Moses at God's dictation.