I am looking for a reference providing an explicit presentation of the map $\mathfrak{C}\Lambda^n_{i}(0,n)\to \mathfrak{C}\Delta^n(0,n)$ for $0<i<n$ (i.e. the inclusion of a cube with its interior and one of its faces removed) as a finite composite of pushouts of simplicial horns. Is anyone aware of its existence in the literature?

@ufabao First of all *as a spectrum * $\mathbb{S}[\mathbb{T}]=\Sigma^\infty(\mathbb{T}_+)=\mathbb{S}\oplus \Sigma \mathbb{S}$, so as a module over $\pi_\ast\mathbb{S}$, $\pi_\ast\mathbb{S}[\mathbb{T}]$ is free of rank 2 on generators of degree 0 and 1 respectively. Let's call them 1 and $d$

Now, $1$ is of course the unit, so to determine the multiplicative structure it is enough to find what $d^2$ is

The multiplication is given by the map $\mathbb{S}[\mathbb{T}]\wedge\mathbb{S}[\mathbb{T}]\cong\mathbb{S}[\mathbb{T}\times\mathbb{T}]\to\mathbb{S}[\mathbb{T}]$ induced by the multiplication on $\mathbb{T}$

Now, if $X$ and $Y$ are spaces, the map $X\times Y\to X\wedge Y$ has a canonical section after one suspension. In particular if $G$ is a group, we can construct a map $\Sigma G\wedge G\to\Sigma (G\times G)\to \Sigma G$

This map when $G=\mathbb{T}$ is exactly the value of $d^2$, pretty much by definition (the section in question is how you split the top cell of $\mathbb{S}[\mathbb{T}\times\mathbb{T}]$), It is a map $S^3\to S^2$ so it can be either $\eta$ or 0 (at least stably). It turns out it's not 0, although I don't know a "nice" proof of this fact

The cofiber of $S^3\to S^2$ is $CP^2$, more or less by construction.

Can you elaborate?

Well, maybe by a slightly different construction. Given $G$, the join $G\star G$ is the pushout of $G\leftarrow G\times G \rightarrow G$, where the maps are projections.

I also have the suspension $SG$, with is the pushout of $*\leftarrow G\rightarrow G$.

Do you mean $\ast \leftarrow G\to \ast$?

The Hopf map is $G\star G\to SG$, induced by the multiplication map $\mu$ on the middle terms.

There's a recipe for relating this to partial realizations of the simplicial object that begins $G\times G \Rightarrow G\rightrightarrows *$.

The realization of the $\leq 2$ part is the cofiber of the Hopf map, which is also $(G\star G)/G$ (group action quotient.

Or you can just note that the Hopf map as I've constructed it is a quasifibration. In particular, all its fibers look like $G$.

In fact, it's a fiber bundle with a little work, I think.

Oh that's nice

Woah, thank you both!

The modern thing would be to use descent to show that the fiber of $G\star G\to SG$ is $G$. You can compute the ring structure on the cohomology of the cofiber directly.

Which is how you show the Hopf map has Hopf invariant 1.

Hi guys! I am struggling at some technical thing about little cube operads

There is an M-little cube operad E_M for M a manifold, defined in this way:

Take BTop(k)^{otimes} to be the nerve of the follwoing topological operad : objects are n \in Fin_*, and maps are Hom(n, m) = U_{\alpha: n \to m} \prod_{i \in m} Emb(R^k \times \alpha^{-1}(i), R^k). In practice it is a relaxing of E_k where we admit all embeddings and not only rectilinear.

It turns out that BTop(k)^{otimes} is operadically equivalent to E_k; infact, if we see rectilinear emeddings of the cube as self embedding of R^n, this inclusion is an approximation.

Now define C_M as the topological category with objects M and R^k and homs such that from M we have always something trivial, and from R^k we always have embeddings; e.g. Hom(M,R^k) = empty, Hom(R^k,M) = Emb(R^k, M).

We can see BTop(k)=Btop(k)^{otimes}_1 inside N(C_M). Take the fiber product B_M = N(C_M)/M \times_N(C_M) BTop(k), and finally set E_M = BTop(k)^{otimes} \times_{BTop(k)^U} B_M^U

If one unwinds the definition, at the topological level, objects of E_M are (f_1, \ldots, f_m) embeddings from R^k to M. Maps from one tuple to the other are multiembeddings that make some diagram commute up to specified (??)

My question is: why is E_{R^k} operadically equivalent to E_k? My route would be to show that the pullback that E_M enters is a homotopy pullback and that B_M^U to Btop(k)^U is an approximation (both are infinity operad with underlying kan complexes, should be the easiest case!). But well, I got stuck in both of them. Any hints on this or possibly other approaches?

Yeah, I know, this is boring :p

For what it's worth, if you wrote stuff using latex (i.e. putting dollar signs around it), I'd be a lot more likely to read it (with ChatJax)

As it stands, it's both a wall of text and a lot of subscripts and superscripts that aren't texed.

Also I'm pretty sure I have no clue how to answer your question, so don't change anything on my account.