Please feel free to delete or move it to somewhere. I just need a confirmation or a reference.

Let $D_r(\mathbb{R}^l,S^n)=F(\mathbb{R}^l,r)_+\wedge_{\Sigma_r}(S^n)^{\wedge r}$ be the $r$-th stable piece in $\Omega^lS^{n+l}$ obtained from Snaith splitting. Is $l(n+r-1)-(r-2)$ the best upper bound on the dimension of $H^*D_r(\mathbb{R}^l,S^n)$ or is there a smaller upper bound? I have used Cohen's computation of integral homology of $H^*F(\mathbb{R}^l,r)$ and the fact that $F(\mathbb{R}^l,r)$ is $\Sigma_r$-equivariantly homeomorphic to $\mathbb{R}^l\times F(\mathbb{R}^{l-1}-\{0\},r-1)$ while I was trying to get some upper bounds. In particular, I am interested in the cases where $r$ is a power of $2$, and homology is with $\mathbb{Z}/2$ coefficients.

EDIT It seems to me that at the prime $2$, the top dimensional class in $H_*D_{2^t}(\mathbb{R}^l,S^n)$ is $$\overbrace{Q_{l-1}\cdots Q_{l-1}}^{t-\textrm{times}}x_n$$ whose dimension determines the homological dimension of the configuration space. I presume, a similar statement would be correct for other primes as well.


Yep, that is the top class, when $r$ is a power of 2, in degree $(2^t-1)(l-1)+ 2^tn$. This also happens to be the top dimension of a cell complex homotopy equivalent to $D_{2^t}(\mathbb R^l, S^n)$: the generalization to all $l$ of the Fox-Neuwirth cell structure shows that there is an equivariant cell complex of dimension $(r-1)(l-1)$ equivalent to $F(\mathbb R^l,r)$.

For other $r$ the top dimensional mod 2 class will be obtained by writing $r$ as a sum of different powers of two, and then multiplying together the associated classes as above.

(Cohen, in SLNM 577, is not fun reading, but homology stuff is in there.)


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