Can a Spiked Gauntlet worn by a polearm-wielding Fighter allow him to do attacks of opportunity with it?
At least one designer assumes, YES
According to the Rules Compendium, pp 5
For example, the rules don’t come out and
say that a Medium creature threatens all squares within
10 feet while wielding a reach weapon and wearing spiked
gauntlets. However, it’s appropriate to assume the creature
does just that.
However, that's from the introduction, under Rules Basics->Adjudication, under a discussion on how
No set of rules can cover every possible
circumstance
And it's qualified as an assumption that's obvious to the designer and author but clearly not to everyone or this question would not have been asked.
Without any other rules source, I would suggest,
It's up to your DM
Let us assume polarm means 2-handed weapon with reach. There are other polearms but they aren't germane.
There are no explicit rules allowing or denying this.
However, if you read the requirement from threatening,
You threaten all squares into which you can make a melee attack
And apply it to the weapon your wielding because
If you’re unarmed, you don’t normally threaten
And extrapolate that threatening is intrinsically tied to the weapon your wielding, then you have to first ask your DM, can I wield more than one weapon with the same hand (or appendage)?
While a spiked gauntlet is a manufactured weapon, and it is implied it can be worn while weilding another (manufactured) weapon, whether or not that means "wielded" as in able to make a melee attack is unclear.
I'm going to suggest No for a few reasons.
While not explicitly stated anywhere, the concept of limiting a hand or appendage to wielding one weapon is littered throughout two-weapon fighting, natural weapons and the equipment entries.
The rules for natural weapons and two-weapon fighting only ever mention wielding one as your primary and one as your off hand or either a manufactured weapon or a natural weapon with a given appendage, but not both. Specifically, the full attack entries for monsters never list the same appendage using more than one weapon (natural or manufactured). Marilith, for example.
This question is about Attacks of Opportunity, not two-weapon fighting or natural weapons, but if you could wield more than one weapon with an appendage, it would show up in those entries.
This concept also follows from this small tidbit from Armor Spikes;
You can’t also make an attack with armor spikes if you have already made an attack with another off-hand weapon
Given that, if your DM decides you cannot wield two weapons at the same time, you have to choose either to threaten adjacent (wielding spiked gauntlets while holding a polearm) or with reach (wearing spiked gauntlets while wielding a polearm). You also cannot switch at the time your enemy provoked because at that point, the decision needs to already be made. It would be akin to allowing someone to threaten with a sheathed weapon because they have quickdraw. It doesn't matter whether or not you can take free actions because it would be out of order. (You have to threaten first).
I don't think it matters much one way or the other how your DM rules, unless you are building a Combat Reflexes based Attack of OppornunityOpportunity build.
Related, Can I full-attack with a Two-Handed Weapon and then attack with a Spiked Gauntlet off-hand?