What does "women come and go "talking of Michangelo" mean in T. S. Eliot's "The Love Song of J. Alfred Prufrock?"
Something quite important, but not previously mentioned, is that this line is repeated twice in Eliot's poem. Noting this element of repetition, let's explore this line further in full context of both it's incidences.
The first incidence is isn't far into the poem:
Let us go then, you and I, ... / ... / To lead you to an overwhelming question ... / Oh, do not ask, "What is it?" / Let us go and make our visit. / In the room the women come and go / Talking of Michelangelo
The omitted few lines truly sets a dreary and dismal scene. This couple is not setting out into happiness. Instead he speaks of "half deserted streets" and "restless nights" and "insidious intent" and "cheap hotels." Their destination (after traversing the yucky urban streets): a room housing some kind of social event. In this room "women come and go / Talking of Michelangelo." Even though I've never heard a critic say it, I can't help thinking of an art gallery during the opening of an artist's most recent show. I love how our eNotes summary say that it is "some sort of tea party or reception, that will feature a great deal of intellectual pretension and inflated talk about art." (Sure enough, later Prufrock talks about taking "toast and tea.") In this respect (and within the context of it possibly being an art show and reception where toast and tea are served), one can imagine women mingling about randomly talking about famous artists of the past, Michelangelo among them. (And I suppose if one of them brought up something avidly interesting about Michelangelo or his works of art, that could circulate the room and become the main topic of conversation for hours.) In this regard, that is what the line in question "means," as your question asks.
The second incidence is a bit further:
And time yet for a hundred indecisions, / And for a hundred visions and revisions, / Before the taking of a taking of a toast and tea. / In the room the women come and go / Talking of Michelangelo. / And indeed there will be time. / To wonder, / "Do I dare?" and, "Do I dare?"
As an element of true repetition, the words are exactly the same: "In the room the women come and go / Talking of Michelangelo." The meaning does not differ from the meaning above. The repetition is simply to emphasize just how boring and repetitive all of this conversation is during this social event. In my opinion, the repetition of these lines proves the boring and humdrum nature of a social event where no one really cares what is being said, pointlessly mingling among themselves while they try to figure out interesting bits of conversation to mention. Here these words are, in the midst of Prufrock's many worries. The context of these lines in the second instance are interesting. Before and after the boring conversation at the social event is the boring time before hand and the anxious time after. The boring amount of time before will allow for "a hundred indecisions" (discussing things but never deciding) and "a hundred visions and revisions" (stating ideas and images, but then changing them again and again). All pointless conversation leading to more pointless conversation at the party.
The anxious time after the lines you mention is Prufrock being anxious about how he will be perceived in a room of beautiful, ditzy women. Will they notice his weight? Will they notice he is balding? Will they notice how he is dressed? What will Prufrock say? He imagines this as well.
Thus, the repeated words you mention "mean" that Prufrock has gone to a social event where beautiful (yet possibly ditzy) women are discussing art. Their conversation does not change which reveals the dismally boring content of the evening.
Lines 13 and 14 of T. S. Eliot's "The Love Song of J. Alfred Prufrock" are as follows: "in the room the women come and go / Talking of Michelangelo." The opening of the poem is quite negative, speaking of "etherized" patients on tables, "half-deserted streets," and "tedious arguments."
The line in question is no different. The speaker, openly angry with life, passes on his anger to the reader because he finds that women spend time speaking superficially about Michelangelo (or art). Given that Michelangelo is a renowned artist, it is very easy to sound knowledgeable about art. Yet, his work does not exist as the epitome of the whole period. Therefore, the conversation about the artist illustrates the superficial nature of the women (and perhaps women as a whole). The knowledge of the women is limited, and the narrator is rightly upset by their superficiality.