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For the homotopy groups of $U(r)$, hence also of $BU(r)$, you can look at the survey by M. Mimura, "Homotopy theory of Lie groups", which is chapter 19 in the "Handbook of Algebraic Topology", edited …

A pretty direct argument was given by Frank Adams in the proof of 16.6 (page 336) in part III of his 1974 Chicago lectures (MR0402720). Thinking of the Atiyah--Hirzebruch spectral sequence for $K^*(X …

An $n$-dimensional CW complex with a single $n$-cell is reducible if the projection $X \to X/X^{(n-1)} = S^n$ onto the top cell admits a section up to homotopy. It is stably reducible, or S-reducible …

The proof of Theorem 9.3.5 (especially the part on page 486) in Spanier's "Algebraic Topology" may be more to your liking. It presumes you have already established the relative Hurewicz theorem, e.g. …

Jean-Louis Loday told me about the extended action by $\Sigma_{j+1}$ in the fall of 1992, after an Oberwolfach talk I gave about the rank filtration of algebraic $K$-theory, where the $\Sigma_j$-repre …

By May's Remark 2.6, the answer should be no. If $X$ is a Moore space with $\tilde H_*(X; Z) = Z/p$ concentrated in an odd degree $2q-1$, and $\tilde H_*(X; Z/p) = Z/p\{x, y\}$ with $\beta_1(y) = x$, …

When $k=2$ and $n$ is even, $$ D_{k,r} S^n \simeq BB_{r+} \wedge S^{rn} $$ is the $rn$-fold unreduced suspension of the classifying space of the braid group $B_r$ on $r$ strings. I once cited Cohen-Ma …

Waldhausen, Jahren and myself proved a fiber gluing lemma for Serre fibrations, in the context of simplicial sets, that may be useful. In Propositions 2.7.10 and 2.7.12 of "Spaces of PL manifolds and …

Question 1: There are several arguments. In degree 2 there is a reference: the proof of corollary 3.7 of Waldhausen's "Algebraic K-theory of spaces, a manifold approach". See http://www.math.uni-bie …

Let $R$ be bounded below, bounded above and nontrivial, and work in $R$-modules. Let $X = \bigvee_{n\in\mathbb{Z}} \Sigma^n R$. Then $X \overset{\simeq}\to D(DX)$ is an equivalence, but $X$ is not d …

The proposed argument for why $Q = S \cup_2 e^1 \cup_\eta e^3$ is not a Thom spectrum seems to use that the Thom isomorphism commutes with the Steenrod operations, which is often false. The deviation …

(A comment to Tyler's answer.) Strictifying pairings from the stable homotopy category to spectra can be tricky. To even get started with an inductive approach let me assume $E$ is connective, so th …

Looping the fiber sequence $S^1 \to S^3 \to S^2$ gives $\Omega^2 S^2 \simeq \mathbb{Z} \times \Omega^2 S^3$. This is the group completion $\mathbb{Z} \times BB_\infty^+$ of $\coprod_{n\ge0} BB_n$, so …

(Not an answer, but long for a comment.) Spanier's "Algebraic Topology", Section 2.7, gives Hurewicz' proof of the theorem that a local (Hurewicz) fibration with respect to a numerable open cover of …

The answer to your third question is "yes". A little more generally, if $Sq^{n+1}$ acts nontrivially in the mod $2$ cohomology of $C$, then $Sq^{(k+1)(n+1)}$ acts nontrivially in the mod $2$ cohomolo …