Books: homotopy groups of inverse limits Can anybody recommend a book, which contains comprehensive information about homotopy groups of inverse limits?
 A: I doubt that there is a book as you desire. There is only a loose relation between homotopy groups of the inverse limit and the homotopy groups of the spaces in the inverse system  which makes it very difficult to obtain  useful results. More interesting are the homotopy pro-groups and the shape groups of spaces. See for eaxmple

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*Mardešic, Sibe, and Jack Segal. Shape theory: the inverse system approach. Elsevier, 1982.

Edited:
In your comment you refer to a paper by Joel M. Cohen. This only considers a very special case: The inverse systems under consideration are inverse sequences $\mathbf X$ having fibrations as bonding maps. As Tyrone comments, the inverse limit of such a system agrees with its homotopy limit. The exact sequence
$$0 \to \lim\nolimits^1 [SY, \mathbf X] \to [Y, \lim \mathbf X] \to \lim [Y, \mathbf X] \to 0$$
is therefore a special case of
$$0 \to \lim\nolimits^1 [SY, \mathbf X] \to [Y, \operatorname{holim} \mathbf X] \to \lim [Y, \mathbf X] \to 0$$
which is valid for all inverse sequences $\mathbf X$.
See for example


*Bousfield, Aldridge Knight, and Daniel Marinus Kan. Homotopy limits, completions and localizations. Vol. 304. Springer Science & Business Media, 1972- Chapter IX


*Edwards, David A., and Harold M. Hastings. Cech and Steenrod homotopy theories with applications to geometric topology. Vol. 542. Springer, 2006 - in particular (5.2.1)
For $Y=S^n$ we obtain the exact sequence
$$0 \to \lim\nolimits^1 \pi_{n+1}(\mathbf X) \to \pi_n(\operatorname{holim} \mathbf X) \to \lim \pi_n(\mathbf X) \to 0$$
The term $\lim \pi_n(\mathbf X)$ is nothing else than the $n$-th shape group of $\lim \mathbf X$ and $\pi_n(\operatorname{holim} \mathbf X)$ is the  $n$-th strong shape shape group (or strong homotopy group) of $\lim \mathbf X$. This has in general nothing to do with $\pi_n(\lim \mathbf X)$.
For the shape groups see 1. and for the strong shape groups see e.g.


*Mardesic, Sibe. Strong shape and homology. Springer Science & Business Media, 2013 - Remark 19.3

As an example for "non-nice behavior" take the Warsaw circle which can be written as the inverse limit of plane annuli $A_n$ (i.e. copies of $S^1 \times I$) bonded by inclusions (i.e. maps of degree $1$). The Warsaw circle is simply connected, but $\lim \pi_1(A_n)  = \mathbb Z$ and $\lim\nolimits^1 \pi_2(A_n) = 0$.
