R. Yosef Shalom Elyashiv recently stated that women are not allowed to wear contemporary wigs (link 1, link 2). Let's take this opportunity to review the literature on the subject and I'll explain why I don't understand the position that forbids wigs.
I. Covering Hair
A married woman has to cover her hair. The Gemara (Berakhos 24a) states:
R. Sheshes said, "A woman's hair is ervah, as it is said, 'Your hair is as a flock of goats' (Song 2:14)."Ervah means that it is considered nakedness and must be covered in public. Therefore, a married woman must cover her hair. Is a wig considered covering? I don't really see how it cannot be considered covering because, factually, it is; the woman's hair is covered.
Click here to read moreII. Early Debate
The Mishnah (Shabbos 64b) states that a woman may go out in public on Shabbos with a wig on her, i.e. her wearing the wig is not considered to be carrying. The Gemara explains that the wig is so the woman looks good for her husband. The Shiltei Gibborim deduces from this Mishnah that wearing a wig is sufficient covering for a married woman's hair. However, others disagree.
The Be'er Sheva (Responsa, no. 18) quotes R. Yehudah Katzenellenbogen who disagrees and offers multiple alternative explanations for the Mishnah, chief among them that the woman is wearing a scarf on top of the wig (see here: link).
The Magen Avraham (75:5) and Shulchan Aruch Ha-Rav (75:4) rule like the Shiltei Gibborim and the Peri Megadim (Esehl Avraham, ad loc.) but others disagree strongly (the Minchas Elazar, in his Nimukei Orach Chaim, suggests that a mistaken student wrote that ruling in the Magen Avraham).
III. Contemporary Debate
R. Ovadiah Yosef (Yabia Omer, vol. 5 Even Ha-Ezer no. 5) argues that the majority of authorities forbid wigs and rules likewise. R. Moshe Feinstein (Iggeros Moshe, Even Ha-Ezer vol. 2 no. 12) argues the opposite, that the majority of authorities permit wigs. He then grapples with important but ancillary issues, such as maris ayin (does it look like the woman is not covering her hair), and says that since it is standard in the community to wear wigs there are no such problems. (See here where R. Yirmiyahu Kaganoff writes that "[m]ost of the great gedolim of North America in the last generation allowed the wearing of [wigs].")
In the Yiddish recording of R. Yosef Shalom Elyashiv, he says that the historical debate does not apply to wigs today (presumably, custom wigs) that are very difficult for onlookers to tell apart from real hair. Therefore, he says, they are ervah.
I don't understand what he means. As a matter of simple fact, the woman's hair is covered The wig is a product of someone else's hair, probably many different people, that serves as clothing that covers the body. I don't see how it can be ervah.
IV. Provocative Attire
I do see why it could be problematic for another reason -- if it is considered provocative clothing. If a person wears a body suit that looks just like a naked body but is entirely synthetic, it is obvious that it is not literally nakedness. However, because it is provocative and will evoke improper thoughts among onlookers, it may not be worn in public. Perhaps some may equate custom wigs with bodysuits. But as a matter of judgment, I find that universal equation difficult because the degree of provocativeness depends on place and time. In today's society in America, a woman's uncovered hair is not particularly provocative. While this does not remove the problem of ervah from a married woman's hair, it makes it hard for me to understand labeling a wig -- any wig -- as provocative.
It is not my place to argue with R. Elyashiv (although my halakhic authorities do not agree with his ruling). However, I can say that I just don't understand what he meant.