What's the story with "undershirt Tzitzis"? In case you've never heard of them, they're special Tzitzis made to be worn directly on the body. So is there any problem with them? Like, do some hold they're no good?
As I understand it, there are two issues:
- Is it appropriate to wear tzitzis directly on your skin, where they can get sweaty?
- If you have a T-shirt-like-shape, but it still has four square corners, is that obligated in tzitis like a four-cornered garment?
As far as what's appropriate, I don't know. Biblically, people just attached fringes to their normal clothing, which was four-cornered, so of course it got sweaty (and I assume was up against their skin). Today that we wear a special garment for the sake of the mitzva, many people like to keep it separate from the skin, but I don't know sources on that (or if it's in fact required anyplace, or just a practice that developed.)
As for the shape, see above. The first Lubavitcher Rebbe in his Shulchan Aruch HaRav says it's okay, in his siddur says not. Chabad-Lubavitch hassidim consider the latter source more authoritative. Non-Lubavitchers often take the Shulchan Aruch HaRav into consideration (especially when it comes to informing non-Lubavitch Hassidic halacha), but I think they treat the siddur as specifically Lubavitch practice. The Chayei Adam (who was non-Hassidic; he lived in Vilna and was strongly influenced by its Gaon) says he sees French Jews attaching fringes to their ordinary vests, which happen to be four-cornered. He thought that was great. I don't know of other sources on this.
The Baal Hatanya, in his Shulchan Aruch (10:18), says that as long as visibly more than half the length is open along both sides of the garment, then it's fine.
However, in Hilchos Tzitzis in his Siddur (which was written later, hence it represents his final decision), he says that it should be completely open on both sides, not connected even with laces, let alone stitches. According to this, then, undershirt tzitzis are no good.
Apparently the Siach Yitchok in Siman Ches says they are no good.
See my update here: May the Tallit Katan be worn directly on the body?
The fact that it is made to absorb sweat and act as an undershirt for this purpose can, according to some interpretations of a ma'amar chazal make it not chayuv in tzitzis alltogether as it is not considered a "beged". Don't have the sources on me but I remember seeing it 20 years ago in a sefer Kol Bo on Hilchos Tzitzis.