If the obstructionist house refuses to act on raise the rate, then there is not much the President can do.

I really have to laugh at the trite little phrase "there is not much the President can do"

And the tax cuts for everyone expire. Then he can re-introduce tax cuts for the middle class and dare the repubs to obstruct that.

Barring any action, the tax rate for every income bracket will revert to pre-2001 levels.

If the Republicans want to present a temporary tax cut for income brackets under $250,000 at that point, the President will be happy to sign it. If not, the tax rate stays at pre-2001 levels by law.

It's all very simple. What does Obama or the Congress have to do for top income bracket tax rates to increase/

Nothing.

and why are the republicans so dead set against it?

Obama campaigned on a policy of deficit reduction using two methods:

1) Cut spending responsibly from wasteful or unnecessary programs
2) Raise government revenue by raising the tax rate modestly on the top income bracket (over $250,000/househol or $200,000 for individual filers).

This was not some "hidden" policy. It was a centerpiece of his campaign. It was in every stump speech. It was in nearly every ad. It was featured prominently in all his campaign material.

And the American people voted for it.

I love the situation, though. The President holds all the cards. Either the Republicans will capitulate to his wishes, an enormous victory for him, or he can simply let the cuts expire and rightfully blame the Republicans for letting it happen. Either way, he'll ultimately get what he wants.