I have a maple tree of which I have poured a cement patio around half of it and I have cut off the water to the tree

The tree is dying as there are only half the leaves there were last year

Also the are cracks in the bark on the trunk extending from the ground , up about 10 feet

Id like to cut it down and hopefully have some suckers start to grow from the trunk

When is the best time to cut it down and is there anything I can do to promote sucker growth

I've got lots of landscaping experience but frankly have never attempted what you are attempting - and don't know of others who have... but I do have some input you might consider:

1. With half of the area around the tree open, I do not believe that the concrete is causing the tree to not get enough water.

2. Even if that IS the case, then you'll have the same problem with the new tree.

3. Maple trees are notorious for shooting out roots close to the surface... and unless yours is a dwarf maple, then eventually the roots may bust up that concrete.

4. I think it more likely that the problem with the tree is only coincidental to the concrete install.

5. I recommend removing the current tree, cutting/pulling out the roots as best you can and planting a new tree that
a. will not grow too big for the area it's in, and
b. is known for deeper roots.