There are a multitude of sellers out there peddling fake gear. Some sellers are careful not to say that the goods are genuine, or original. Others are a little more outlandish with their claims.
Some people may be happy to pay less for a fake Lacoste or Nike TN item. The issue comes when people pay good money for what turns out to be a fake. There are a few telltale signs which will help you to decide.

Firstly, Lacoste do not make hooded tracksuits. Neither to my knowledge to Nike TN.  A few months ago there was a plethora of such fake tracksuits being sold on eBay. The design was much the same for both the fake Lacoste and the fake TNs. I bought a fake Lacoste tracksuit to see what it looked like.
Generally, the fake hooded tracksuits  look like this:



In this case, the badges and labels are Lacoste. However, almost exactly the same tracksuit can be found with TN badges and labels. The picture below shows a fake TN tracksuit in exactly the same design but a different colour:



Another popular design is the one shown below:



Again, this design recurs on eBay. Lacoste have never made such a design.
Another key to spotting a fake Lacoste is the sizing. Lacoste always size their tracksuits by number. If a seller is advertising XL or L and doesn't mention a number then it is probably a fake. If in doubt, email the seller and ask what the size says in the label.

As for Airmax, there are also a large number of fakes out there. These are often more difficult to spot. A popular design is the one show below:



The trouble is that they look exactly like the genuine article in the pictures. However, in the flesh there are crucial differences, not least regarding quality!
Firstly, ask the seller if the drawstring on the bottoms is on the inside or outside of the waist. They are always on the inside for the genuine article.
Second, the colour of the inside of the tracksuit is important. A geniune tracksuit, being of better quality, feels better. I have a pair of genuine dark grey bottoms and the inside is a much lighter grey and feels almost woollen. Fakes will probably tend to be the same colour as the outside.
Do not be fooled by labels! Labels are the easiest thing of all to copy. A genuine tracksuit will have a loop inside at the back (just like you get on a jacket for hanging it on a hook) with AIRMAX written on it followed by the tick. Beneath this will be the sizing label and just next to it the big label with material and Keep Away from Fire in various languages. A fake tracksuit may have the labels in a different position, and they will also not be as sharp as the real thing. Often the labels will also be thinner.

The upshot is that if something seems too cheap to be true, then it usually is. If you don't mind wearing fake gear, then fine. However, the quality is always far below that of the genuine products and you will probably find that after a few washes the stitching will become loose.