How we cite our quotes:
Troy: "I don't know why [Lyons] don't go and get him a decent job and take care of that woman he got." (1.1.168)
Troy sees it as a man's duty to take care of his woman. The fact that Lyons lives off Bonnie's income is disgraceful to Troy. In Troy's mind, Lyons is failing in his duty as a man. However, we wonder if Troy feels that he failed in his duty as a father by not being around to provide for Lyons when he was a child.
Rose: "Can't nobody say you ain't done what was right by him [Gabriel]." (1.2.67)
Here Rose tries to get her husband to see that he's fulfilled his duty to his brother. It seems Troy is concerned that people will think he took advantage of Gabriel by using his disability checks to buy a house and then kicking him out. This doesn't seem to be the case, however. Troy is definitely flawed in many ways, but he does seem to feel a sense of duty to his family.
Rose: "He [Troy] say you were supposed to help him with this fence."
Cory: "He been saying that the last four or five Saturdays, and then he don't never do nothing, but go down to Taylors'." (1.3.4-1.3.5)
There's a good chance that all those times Troy said he was going down to Taylors' he was really going to Alberta's house. It seems to us that in some ways he's been shirking his duty to his family. The half-finished fence in the yard could be seen as a symbol of Troy's failure to uphold his familial duties.