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Let's be precise about the question! I claim it is not meaningful until you choose basepoints in X and Y and restrict to based maps, since otherwise the suspension used to define the stable homotopy …

This is an excellent question that I have thought a lot about. I'd rather answer it in a more general context that was motivated by what I knew to be true in the stable homotopy category. The referen …

An axiomatization of exactly how smash products of cofiber sequences should behave is given in the context of triangulated categories in my paper The additivity of traces in triangulated categories …

Sure. Smashing a based space with a spectrum is equivalent to smashing its suspension spectrum with that spectrum. So it suffices to give a nontrivial space whose reduced $K$-homology is trivial. …

This is a comment, not an answer, I suppose. Just a reference to Adams and Walker "On complex Stiefel manifolds''. This follow up to Adams' "Vector fields on spheres'' directly computes the $KO$-gr …

In 2002, Paul Balmer wrote a nice two page note answering the same question. He sent it to me because I had asked the same question in my paper ``The additivity of traces in triangulated categories'' …

I didn't want to answer this because the question seemed too elementary to spend time on. But to see quasicategories invoked for something so classically elementary is truly painful. (Forgive me Dyl …

Actually, this is a subtle and interesting question, and it is to some extent model dependent. When one defines the integer graded homotopy groups, one does so using colimits over representation spher …

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 …

ps: I really don't like ``if you really must ...''. There are serious advantages to working in a model category in which every object is fibrant, and, related to that, both for theory and computatio …

This is probably belaboring the obvious, but just take seriously the equivalence between grouplike $E_{\infty}$ spaces and connective spectra. See for example Equivalence between $E_\infty$-spaces …

When I wrote the comment above, my memory was blanking. The connection between ring maps $MU\to R$ and complex orientations that Mark describes goes back to Quillen's original work relating $MU$ to fo …

I have nothing non-trivial and non-digressive to say, but it might help to point out in an elementary way some things that may be relevant. One way to think about things is that there are distinction …

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 answer to this question is in LMS (I.6.9 of http://www.math.uchicago.edu/~may/BOOKS/equi.pdf) and in McClure's contribution to BMMS (VII\S1 of http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf), …