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I'll give a shot at an answer. The relevant dimensions are of the form $2^j-2$. For $j\leq 4$, it is easy and classical that we can construct manifolds of Kervaire invariant one. The problem was ``r …

Here is a fact that should be much more widely known than it is. The category of posets is isomorphic (not just equivalent) to the category of $T_0$ Alexandrof spaces. A topological space is said to …

The homotopy analogue definition of natural transformations has been known and used regularly since at least the late 1960's, by which time it was understood that the classifying space functor from (s …

The subject is really way too big (as are so many others of course). I worry a lot about students not in Cambridge or Chicago or Stanford or other places where there are people with folklore at their …

I suppose I should try to answer since the question of whether or not $BP$ is an $E_{\infty}$ ring spectrum was Problem 1 of "Problems in infinite loop space theory'', http://www.math.uchicago.edu/~ma …

Working simplicially (in those days called "semi-simplicially") this is surely due to Moore, with details in unpublished 1956 lecture notes and in John C. Moore, Semi-simplicial complexes and Postniko …

There is a short classical paper that has not been mentioned: Boardman, J. M. The principle of signs. Enseignement Math. (2) 12 1966 191–194. For products in homology and cohomology theories, there …

Since G is finite, there is no problem with just repeating the proof in the case $G=e$, using $Z$-graded homotopy group functors on the orbit category. Take $D^{\leq n}$ to be the spectra whose homot …

The equivalence (P) is a deep and subtle property of the smash product of spectra in modern symmetric monoidal models for the stable homotopy category. It is very unlikely to hold in other contexts. …

This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal category is called a permutative category. I observed ages ago (http://www.math.uchi …

The late Gaunce Lewis's 1978 PhD thesis ``The stable category and generalized Thom spectra'' proved (as a special case) that the Thom spectra of $F$ and its oriented version $SF$ (alias $GL_1(S)$ or $ …

my friend, I have an email! But I can offer the history. First, although the $E_{\infty}$ book was published in 1977, it is a shotgun marriage of a bunch of earlier preprints that were rejected for …

Chris, that is not actually what we did. Personally, I find indexing simplicial sets by inner product spaces to be unnecessary and unhelpful, and I've not coauthored any paper with such a constructi …

Oh, come on! Prop. 17.4, p. 69, of my ancient but still current book ``Simplicial objects in algebraic topology'' proves that the homology groups of the Moore complex of a simplicial group $G$ are t …

There is a very careful analysis of this question in Lemma 3.4.2, page 57, of More Concise Algebraic Topology, by Kate Ponto and myself. Assuming that $E$ and $B$ are connected, a fibration $E\longrig …