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As stated in previous answers, the answer to the question is no. Here is another viewpoint, which gives additional information about the relationship between $G$ and $BG$: A pair $(G,BG)$ where $G$ …

Hej Craig, Re (2) as Tilman says in his comment, there is an unstable homotopy fixed point spectral sequence, a special case of the spectral sequence of a homotopy limit as described by Bousfield and …

No, this is in general quite far from being true (and also somewhat ill-posed, since there is not a natural candidate for what "=" means since there is no map). For example, take $M = S^1$ and $N=S^2 …

Even for $n=1$ the cofiber will almost never be a loop space. There is no reason for it to be a loop space, and plenty of reasons for it not to be a loop space. E.g., it would need to be a simple s …

If you are willing to make "niceness" assumptions, assuming that the n-type is, say, nilpotent with finite fundamental group, then the dimension, in any reasonable sense, will always be infinite, sinc …

The space $BG/G$, with the model you describe, has been studied, but is not completely understood. It's $\pi_1$ identifies with the abelianization, but there may be higher homotopy. It identifies wi …

Centralizers of various types of subgroups, and their relationship to invariants like the cohomology of the classifying space $BG$ has a long history of study, at least going back to the papers from t …

I'm not sure exactly what you are after, but here is an elementary discussion, surely well known to you. Rephrasing what Mark Hovey said, the first question you should ask yourself is probably why loc …

Repeating Mark Grant's comment, the spectral sequence when all spaces are $K(\pi,1)$s goes under the name Lyndon-Hochschild-Serre spectral sequence. Good references for this spectral sequence are: …

A good reference for 1) is Bourbaki: Lie groups and Lie algebras Chapter 9. See in particular Section 4.6. In particular it follows that 2) $\pi_1(G/T) = 0$ and that 4) $\pi_1(G)$ is finite if and o …

The problem you mention has a long history. The best homotopy characterization is probably using the notion of finite loop spaces: A finite loop space is a space $BG$ such that $\Omega BG$ is homotopy …

Yes: As has been pointed out, admitting the structure of a connected $A_\infty$-space is the same thing as being homotopy equivalent to a connected loop space. The Hilton-Roitberg criminal, mentione …