Are there any cities that are (significantly) more conservative than their suburbs?
Alternatively: are there any suburbs that are significantly more left-wing than the urban area?
Technically, yes, a lot of inner city voters aren't exactly "liberal" but poor black/Hispanic voters who vote Democratic because of welfare and social programs.
So, liberal.
A lot of them don't really care for social issues like abortion or LGBT rights.
If your definition of "don't really care" is that they oppose abortion OR LGBT rights then your really wrong.
if your definition of "don't really care" actually means they dont care much about abortion or LGBT rights then your probably right.
Which leaves economic policy as the reason they vote democrat.