Do you mean tzitzis on a mesh garment? I don't see the problem, why should we care what kind of weave it is? The majority of coverage is cloth not air.
As for synthetics, here's a piece from Rabbi Heinemann's Star-K article:
... it is the opinion of a leading Torah authority that garments woven with synthetic fibers do not require tzitzis. This means that a polyester or rayon garment does not require tzitzis. What about a wool-rayon blend, which is a blend of natural and synthetic fibers? The criteria for determining tzitzis obligation is which of the two fabrics covers the majority of fabric area. If, for example, the fabric is a 50-50 blend of natural and synthetic fibers, but the synthetic thread is thicker than the natural, then the garment would not require tzitzis. Wearing an Arba Kanfos (beged with tzitzis) made of synthetic fabric would present the problems stated earlier: 1) non-fulfillment of the mitzvah, 2) uttering a blessing in vain, 3) carrying on Shabbos.
I think most of the cotton-poly blends I've seen are something like 75% cotton, so they should be the same halachically as 100% cotton. (Is polyester thread usually thicker than cotton thread? Anyone here a textile maven?)
R' Moshe Feinstein's responsum concerning cotton tzitzis addresses the question of synthetics as well; if I recall, he points to the Rambam who says there's no mitzva of tzitzis on leather garments. Usually a leather garment isn't woven; now you could cut strands of leather and weave them, but I think R' Moshe says it's likely that because leather starts off as a sheet and not fibers, there's no mitzva even if you do weave it. I think R' Moshe argues the same for synthetics (don't recall if he leaves it as a question, or concludes stringently); if R' Moshe is stringent, that's most likely the "leading Torah authority" cited by R' Heinemann.