Is it better halachikally (according to jewish law) to have the beged (garment) of your tzitzis to be made from wool as opposed to cotton? Is it a chumrah?
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There are opinions (in particular, the Rambam, Laws of Tzitzis 3:1-2) that only woolen or linen garments are required by Torah law to have tzitzis, and that all other fabrics are obligated in tzitzis only Rabbinically.
Also, there is another opinion (cited in Hagahos Maimonios ibid. 3:6) that garments of other fabrics are indeed required to have tzitzis by Torah law, but that only tzitzis of combined wool and linen (which normally would be shaatnez, but which the Torah permits for purposes of tzitzis - Rambam ibid.) are acceptable.
So according to those opinions, it is indeed better halachically to wear a woolen beged with woolen tzitzis, because then you avoid all of these uncertainties, and can be sure of fulfilling the mitzvah on a Biblical level.
The Beit Yosef (OC 9), the great collector of Rishonim's opinions, cites two opinions that require Tzitzit by Torah law only on wool and linen: the Rambam (Tzitzit 3:1) and the Rif (Tzitzit 14a). He then cites Rashi, Rabbeinu Tam, Ri, Mordechai, Semag (ie a lot of Ashkenazi Rishonim), Tur and Rosh who wonders how the former two opinions could rule that way, based on certain technical details in the relavent piece of Talmud (Menachot 39b). The Beit Yosef however, is not interested in arguments and blindly follows his rule of ruling like the majority of Rambam, Rif and Rosh.
The Darkei Moshe (Rama's commentary to the Tur) there complains that this is not how Halacha is supposed to work, and cites his introduction to Darkei Moshe where he complains about the Beit Yosef's methodology. The Darkei Moshe follows the majority opinion in Rishonim (certainly the majority of Ashkenazim) and common Ashkenazi practice in ruling that all materials are obligated by Torah Law. He also cites the Haghot Maimoniot (another Ashkenazi Rishon) who rules similarly and cites Maharam of Rothenburg and the Sefer HaTerumah who agree. The Haghot Mainoniot mentions an opinion of Rabbeinu Simcha cited in the Yere'im who rules that it is a doubt and one should be stringent to wear wool, but he doesn't accept that ruling and neither does the Darkei Moshe. [Also, it is quite rare that we rule like the Yerei'm.]
Accordingly, it should come as no surprise that major Ashkenazi Achronim rule like the Rama (including the Bach, Gra and Shulchan Aruch haRav to OC 9). Cotton Tzitzit were worn daily by both the Chazon Ish (source) and Rav Joseph Soloveitchik (source).
So if you are Sephardi, then there is certainly a preference to wear wool Tzitzit. If you are Ashkenazi, there is very strong basis not to have such a preference.
Generally it's Sephardic Jews who are more concerned for the opinion of the Rambam. For Ashkenazim, both the Mishnah Brurah and Rabbi Moshe Feinstein consider it a meritorious chumrah, though the former uses stronger language. (The latter acknowledges that it can be hot in American summertimes!)