15 votes
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Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney ...
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6 votes

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

Put $$ R(n)=H^*((S^2)^{\times n}) = \mathbb{Z}[x_1,\dotsc,x_n]/(x_1^2,\dotsc,x_n^2) $$ A key point is that if $u\in R(n)$ with $|u|=2$ then $u^2=0$ iff $u=0$ or $u=m\,x_i$ for some $m\in\mathbb{Z}\...
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5 votes

Relation between de Rham cohomology group of Lie group as a manifold and group cohomology of Lie group

The answer to the question is: not really. Consider $G=\mathbb{R}^n\,.$ This has nontrivial group cohomology in all degrees $\le n$ (by the van Est isomorphism theorem, for example), while the ...
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4 votes
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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 ...
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4 votes

What is $TP(\mathbb{Z}_p)$?

The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by ...
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2 votes

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 ...
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