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It is not. See Theorem 1.4 of this paper by I.M. James (Trans. AMS 84 (1957), 545-558). In particular, there exists no homotopy associative multiplication on $S^n$ unless $n=1$ or $n=3$.

Okay, so maybe I'm being naive here, but I think this is probably a naive question. The answer might be a little complicated by the fact that I don't have any good way to drawcommutative diagrams on h …

This is sort of a two part question: 1) In one construction of $BP$, the Brown-Peterson spectrum, one uses this idempotent map of Quillen's, $g:MU_{(p)}\to MU_{(p)}$ and from what I can tell, the ima …

So it seems to me that, using Peter May's paper above, for two distinguished triangles $X\to Y\to Z$ and $X'\to Y'\to Z'$, we have a 'filtration': $$Z\wedge Z'\overset{f_1}\leftarrow Y\wedge Y'\overs …

There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of th …

Is it true that the Moore spectrum for the group $\mathbb{Z}_{(p)}$ can be constructed by smashing $\mathbb{S}$ with $q^{-1}\mathbb{S}$ for each $q\neq p$ (here both $q$ and $p$ are primes). It seems …

In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems …

I have seen some similar questions to this one on here recently, so I hope this isn't redundant. Basically, suppose I have two cofiber sequences of spectra (or perhaps just work in some general homot …

This question is specifically related to the spectra $X(n)$ used in Devinatz, Hopkins and Smith's proof of the nilpotence conjectures, but any general answer in terms of the Thom isomorphism would als …

In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is equiv …

Suppose I am given a morphism $f:BG\to BGL_1(R)$ for $R$ some at least $E_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism $G …

So this can definitely be done. It took me a while to figure out all the details, but in the end it's not so conceptually complex. The basic idea is that if you've got a fibration $F\overset{i}\to E …

This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!). If I have a d …

Let $E$ be a spectrum having exactly two non-trivial homotopy groups, $\pi_k(E)=G$ and $\pi_j(E)=G'$ for $j>k\geq 0$, and having $k$-invariant $\alpha\colon\Sigma^{k}HG\to\Sigma^jHG'$. Also assume tha …

Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following: An ideal $J\subset S$ is invariant under the action of the group $\mathrm{ …