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This is Theorem 4.8 in Chapter XIII of Whitehead, George W. Elements of homotopy theory. Graduate Texts in Mathematics, 61. Springer-Verlag, New York-Berlin, 1978. xxi+744 pp. ISBN: 0-387-90336-4 an …

An informal reference could be the diagram on page 2 of the lecture notes from my September 2000 lecture in Oberwolfach, where I discussed the chromatic red-shift conjecture.

(Edited per request to add more detail.) Consider first the case $G = U(n)$. If $S^1 \to U(1)^n \subset U(n)$ is injective, with $H^*(BS^1) = Z[t]$, $H^*(BU(1)^n) = Z[t_1, \dots, t_n]$, $H^*(BU(n)) = …

If you are willing to read a little French, look at page 434 of Serre's "Homologie singuliere des espaces fibres" (1951). For an exercise in English, with a hint, try Exercise 2 on page 155 in Mosher …

A published reference for the claim (right after the question in boldface) is the proof given by Thomason on pages 1657-1658 of "First quadrant spectral sequences in algebraic K-theory via homotopy co …

Some examples with one nonzero family of differentials: The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p= …

The $D$ with $D(xy) = xD(y) + D(x)y$ are the primitives in the Steenrod algebra $A$, which are dual to the indecomposables $\xi_i$ in $A_* = F_2[\xi_i \mid i\ge1]$, so there is one such $D$ in each de …

The initial example of such an $E$ is the Thom spectrum $M\xi$ associated to the $E_1$-map $\Omega \Sigma BU(1) \to BU$, studied by Baker and Richter in "Quasisymmetric functions from a topological po …

(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 map $i : * = B\{e\} \to BString$ over $BF = BGL_1(S)$ is $7$-connected, so induces a $7$-connected map $S = M\{e\} \to MString$ of Thom spectra, by the Thom isomorphism and Hurewicz theorem. At …

You can learn about this, and more, in Part II of Adams' "Stable Homotopy and Generalised Homology" (1974), especially section 4, in the special case $E = KU$.

If you instead work with a cofiber sequence of filtered spectra, then I gave a sufficient condition in Proposition 5.4 of https://www.mn.uio.no/math/personer/vit/rognes/papers/highfix.pdf (JPAA, 1999) …

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. …

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 …

An elementary answer to the first part of your question: Finite sets are more fundamental than their cardinalities. Consider the category of finite sets and bijective functions. Its geometric realiza …