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Do you want to know the integral homology? If you are happy with homology with coefficients in $\mathbb{F}_p$, the best way to compute (and describe) the homology of Eilenberg-MacLane spaces is the t …

Do you mean the argument in the last page of Selick's paper? If so, he does not make use of an "automorphism". What he wanted to say is the difference between the map $$(H'\circ\Omega\gamma)_* : H_*( …

Otogonbayar Uuye observed in 1011.2926 that the category of $C^*$-algebras can be made into a category with fibrant objects in the sense of K.S. Brown in several ways. But there don't seem to be corr …

I don't know who first proved this fact, but a standard reference would be this paper of Toda's. It is stated as Theorem 4.1. Toda's proof is essentially the same as Tom's answer.

I recommend a lecture note by Fred Cohen for the EHP sequence, although it doesn't explain the J homomorphism.

Ralph Cohen, John Jones, and Graeme Segal found an interesting "construction" of a topological acyclic category $C(f)$ from a Morse-Smale function $f : M\to \mathbb{R}$ in a preprint in early 90's. Ob …

Let me add one more introductory book. Hansen, Vagn Lundsgaard. Braids and coverings: selected topics. With appendices by Lars Gæde and Hugh R. Morton. London Mathematical Society Student Texts, 18. …

There is a geometric (space level) way of realizing Eric's answer. In fact, this is the subject of section 1 in a paper by Ravenel and Wilson (MathSciNet). They used the iterated simplcial bar constru …

The following example appears in the definition of twisted $K$-theory. Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$. Since the unitary group $U(H)$ is contractible, t …