You need to make a fresh cut across the top but since you can't drill several holes in it and pour full strength concentrated Roundup in them several times after they drain. That size of a tree you should be able to drill a dozen holes with a large bit.
I killed an apple that way recommended by a nursery.
It probably won't kill the suckers they have probably developed there own roots by now.

Or use stump remover but then nothing around there can be planted for about 2 years and it might kill your grass around it.

No expert here but we had the same problem. When it was taken down, the cut roots sent up suckers for many months, maybe more than a year. We treated them like weeds, pulling or cutting them off. The roots eventually died. No more tree. No answers here for neighbors letting them get out of hand unless you can go tree hunting with a hatchet in their yards too. The stump killer idea will likely work too. The stump was ground off here when they took it out. Linda

You have described the reason why Liguidambar should not be planted in a confined area where surface roots and suckering roots can be a problem. Any tree likely to regrow from the roots should be killed when the tree is first cut. You will have only about 6 hours after cutting the tree down to paint the cut area with 41% roundup straight from the container. If done soon enough the tree will absorb it and it will kill the roots. If you wait until the next morning the tree will have sealed the cut area and it will have no effect. Al

Thanks for all the input. It's only one neighbor that lets the suckers get to 6' before cutting them down. I've been variously pulling them up or hitting them with Roundup at 3-4". I was thinking that the guy letting them get to 6' would have them developing their own roots and spreading the mess, so thanks for confirming that. I'll keep taking them out, and when I can, try drilling holes into the stump and pouring in concentrated Roundup or Tordon.

The temp must be 68 degrees for Roundup to work so don't wait too long this fall. Hit it soon while its still growing good...or trying to anyway.
Make the holes about 3 inches deep so deep enough for fresh wood to absorb it.

I'll be rooting for you to kill those suckers. mbrowne. I am battling Locust tree root suckers and I feel for you.
Renee

I was told to always use Roundup diluted as the package directions state, not at full strength.

The plant must be able to absorb the Roundup, and it cannot absorb it as well full-strength as diluted. The guy said that the manufacturer would be more than happy to have you use it at full-strength, because they sell more that way, but that it is not so effective.

I don't know if that is true or not, but it works on trash-tree seedlings at diluted strength for me, so I dilute.

I love Liquidambar trees as long as they are a few blocks away!

Hoovb To kill the roots of a cut tree the cut surface must be painted, not sprayed, and the Roundup must not be diluted, but be at maximum strength available, which is 41%. This is not a recommendation from the manufacturer, but from an Arborist who I consulted about a tree removal in a location which would not accommodate a stump grinder. That was 30 years ago and I have used it many times here on my six acres of native forest, where I would not have ANY sunny areas for a garden, without regular tree removal. Al