I’ve been saying here that Obama’s reelection chances could rest on his ability to fight Mitt Romney to a draw on the economy, in order to win the election in other areas — entitlements, personal attributes, leadership qualities, who really has the middle class’s interests at heart, etc.

Today’s Associated Press poll sheds more light on this. It finds the two candidates roughly tied on who can be trusted to handle the economy; AP polling has found this for months. But the new poll also finds:

* Fifty-four percent trust Obama to handle social issues such as abortion or gay marriage, versus only 33 percent who trust Romney.

* Forty-nine percent trust Obama to handle Medicare, versus 41 percent who trust Romney. (You might argue that this should be larger for Obama; but that’s still an eight point lead on an issue that is now central to the presidential race, now that Paul Ryan has been tapped.)

* Fifty-two percent say Obama is a strong leader, versus 40 percent who say that about Romney.

* Fifty two percent say Obama says what he really believes, versus 35 percent who say that about Romney.

* Fifty three percent say Obama understands the problems of people like them, versus only 35 percent who say Romney does.

This helps explain, as Jamelle Bouie noted yesterday, why Obama has proven the leading culture warrior in this election, coming out for gay marriage, vowing to protect Planned Parenthood, and running multiple ads highlighting Romney’s and Ryan’s stances on abortion, contraception, and women’s health. Statements such as those from Todd Akin have helped Obama by elevating social issues and giving Dems an easy way to cast the GOP as hidebound and reactionary on them. But Obama himself has very deliberately sought to make this election about these issues, too, in hopes that the already-large gender gap will continue to widen. And by the way, focusing on women’s health is also meant to exacerbate the gap in personal attributes; the Dems’ constant refrain is that Romney’s backward views on the topic show he’s “out of touch.” Meanwhile, Romney has made Medicare central to this election by picking Ryan.

The economy could very well still trump all, particularly if there’s another downturn. But if the Obama camp can neutralize the advantage Romney should be enjoying on the economy — by persuading voters Romney doesn’t have the answers to their economic problems and that the recovery, while too slow and painful, could be upended by a change in policies and leadership — perhaps more of this election will unfold on all this other turf.