# Tag Info

### Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative ...
• 29k
Accepted

### Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory

This is an elaboration on my comment above. Let us consider the natural transformation induced by the diagonal $$\Sigma^\infty\Delta\colon \Sigma^\infty X \to \Sigma^\infty X\wedge X.$$ The natural ...
• 9,649

### Spaces homotopy dominated by $S^2 \times S^2\times S^2$

You're actually right. The cohomology algebra of $X=S^2\times S^2\times S^2$ with coefficients in $\mathbb{Z}$ is $H^*(X)=\mathbb{Z}[x,y,z]/(x^2,y^2,z^2)$ with $|x|=|y|=|z|=2$. The cohomology algebra ...
• 14.1k
Accepted

• 48.6k

### What are Koszul dualities?

I have finally found a source which puts together the pieces in a satisfactory way, at least in the stable setting, here: Amabel, Araminta. "Poincaré/Koszul Duality for General Operads." ...
Accepted

### Maps that preserve winding numbers

I found an answer to this question: the question of how a continuous function changes the winding number of a closed curve can be studied quite generally. The important concept here is the degree of ...
• 1,109
Let me write $S^n/p$ for the cofibre of $p$ times the identity map on $S^n$, or in other words the mod $p$ Moore space with homology in degree $n$. If $p$ and $q$ are coprime then $S^n/p\wedge S^m/q$ ...