I'm following Arone's lectures on Goodwillie calculus from Munster 2015. There he left an exercise:



> 
> Find $\partial\_kF$ for $F: \text{Top}\_\* \to \text{Sp}$ given by $F(X) = \Sigma^\infty X^{\wedge n}/\Sigma\_n$ where $X^{\wedge n} = X \wedge \cdots \wedge X$ and $\Sigma\_n$ is the symmetric group, which acts on $X^{\wedge n}$ by permuting factors.
> 
> 
> 


In the lectures he solves the case $n = 2$, that gives $\partial\_1 F \simeq \Sigma^\infty\Sigma\mathbb{RP}^\infty$, $\partial\_2F \simeq \Sigma^\infty$ and $\partial\_nF \simeq \*$.


I do not know how to work out the general case. I know that $\partial\_kF \simeq \text{hocolim }\Omega^{km}\text{cr}\_kF(S^m, \dots, S^m)$.


For $k = 1$ then $\partial\_1F \simeq \text{hocolim }\Omega^mF(S^m)$. My idea is to write $F(S^m) = S^m \wedge G(S^m)$ for some $G$ in order to cancel $\Omega^m$. This can be done by choosing a $\phi: (S^m)^{\wedge n} \to (S^m)^{\wedge (n-1)}$ such that $(x\_1, \dots, x\_n) \mapsto (x\_1 + \cdots + x\_m, \phi(x\_1, \dots, x\_m))$ is a homemorphism. In this case $(S^m)^{\wedge n}/\Sigma\_n \cong S^m \wedge (S^m)^{\wedge (n-1)}/\Sigma\_n$ where the $\Sigma\_n$-action on $(S^m)^{\wedge (n-1)}$ depends on $\phi$. Unfortunately, I was not able to find a $\phi$ such that this action looks nice or familiar as in the case $n = 2$.


Alternatively, I tought $(S^m)\text{^}^n/\Sigma\_n \cong \text{subset}\_{\leq n}(\mathbb{R^m})^+ \cong S^m \wedge \text{subset}\_{\leq n, 0}(\mathbb{R^m})^+$
where $(\cdots)^+$ means compactification, $\text{subset}\_{\leq n}(\cdots)$ is the space of subsets of cardinality $\leq n$ and $\text{subset}\_{\leq n, 0}(\mathbb{R^m})$ is the space of those with barycenter 0. Then $\partial\_1F \simeq \Sigma^\infty\text{subset}\_{\leq n, 0}(\mathbb{R^\infty})$. This agrees with the previous result for $n = 2$ since any two unordered points can be identified by the line joining them ($\mathbb{RP}^\infty$) and the distance from 0 ($\Sigma$, after compactification). But


1. It does not seem to be the "right solution".
2. What about $\partial\_nF$ for $n > 1$? I do not know how to proceed.


Thanks in advance!


[[original post](https://math.stackexchange.com/questions/3897916/goodwillie-derivatives-of-x-mapsto-sigma-infty-x-textn-sigma-n)]



Goodwillie derivatives of $X \mapsto \Sigma^\infty X^{\wedge n}/\Sigma_n$
Your solution is correct as far as it goes. Let $\rho^{n-1}\cong {\mathbb R}^{n-1}$ denote the reduced standard representation of $\Sigma\_n$. One way to define $\rho^{n-1}$ explicitly is to say that it is the orthogonal complement of the diagonal in $\mathbb R^n$. Let's write it out:
$$
\rho^{n-1}=\{(x\_1, \ldots, x\_n)\in\mathbb R^n\mid x\_1+\cdots+x\_n=0\}
$$
Let $\hat S^{n-1}$ denote the one-point compactification of $\rho^{n-1}$. This is a sphere of dimension $n-1$ with a specified action of $\Sigma\_n$. More generally, for any $m\ge 0$, let us write $\hat S^{(n-1)m}$ for $(\hat S^{n-1})^{\wedge m}$.


Furthermore, $\hat S^{(n-1)m}$ is the one-point compactification of the representation $(\rho^{n-1})^m$. It follows that $\hat S^{(n-1)m}$ is homeomorphic to the unreduced suspension of the unit sphere in $(\rho^{n-1})^m$. Let me denote the unit sphere by $\tilde S^{(n-1)m-1}$. We have a $\Sigma\_n$-equivariant homeomorphism $$\hat S^{(n-1)m}\cong \Sigma \tilde S^{(n-1)m-1},$$ where $\Sigma$ indicates unreduced suspension.


Now let us take colimit as $m\to \infty$. We obtain the following equivalence, where both sides are just notation for the appropriate colimits
$$\hat S^{(n-1)\infty}\cong \Sigma \tilde S^{(n-1)\infty-1}.$$


Let $F\_n(X)=\Sigma^\infty X^{\wedge n}\_{\Sigma\_n}$. By your calculation, $$\partial\_1(F\_n)\simeq \Sigma^\infty \Sigma {\tilde S^{(n-1)\infty-1}}/\_{\Sigma\_n}.$$


To analyze this further, let us calculate the fixed point set ${\tilde S^{(n-1)\infty-1}}^H$, where $H\subset \Sigma\_n$. The following lemma is a quite easy exercise


**Lemma:** Let $H$ be a subgroup of $\Sigma\_n$. Define $o(H)$ to be the number of orbits of the action of $H$ on $\{1, \ldots, n\}$. There is a homeomorphism $${\tilde S^{(n-1)-1}}^H\cong S^{o(H)-2}.$$
Note that if $H$ is a transitive subgroup of $\Sigma\_n$, then $o(H)=1$ and the right hand side of this equation is $S^{-1}$, which is taken to be the empty set.
More generally for $m\ge 1$, there is a homeomorphism
$${\tilde S^{(n-1)m-1}}^H\cong S^{(o(H)-1)m-1}.$$
Taking limit as $m\to \infty$ we obtain the following


**Corollary** $(\tilde S^{(n-1)\infty-1})^H$ is contractible if $H$ is non-transitive and is empty if $H$ is transitive.


This means that $\tilde S^{(n-1)\infty-1}$ us a universal space for the family of non-transitive subgroups of $\Sigma\_n$. Let us therefore introduce the notation $E\_{\mathrm{nt}}\Sigma\_n=\tilde S^{(n-1)\infty-1}$ and $B\_{\mathrm{nt}}\Sigma\_n={E\_{\mathrm{nt}}\Sigma\_n}/\_{\Sigma\_n}$. $B\_{\mathrm{nt}}\Sigma\_n$ is sometimes called the classifying space for the familty of non-transitive subgroups of $\Sigma\_n$. We have proved the following


$$
\partial\_1 F\_n \simeq \Sigma^\infty \Sigma B\_{\mathrm{nt}}\Sigma\_n
$$


Note that when $n=2$, the family of non-transitive subgroups of $\Sigma\_2$ consists of just the trivial group. Because of this, $B\_{\mathrm{nt}}\Sigma\_2=B\Sigma\_2=\mathbb R P^\infty$. So we get the expected answer for $n=2$.


But there is another interesting way to think of the spectra $\partial\_1 F\_n$. Let $Sp^n(X)=X^n/\_{\Sigma\_n}$, where $X^n$ denotes $n$-fold *cartesian* product of $X$ with itself. It follows that for a pointed space $X$, $X^{\wedge n}/\_{\Sigma\_n}\cong Sp^n(X)/Sp^{n-1}(X)$. The construction $Sp$ extends from spaces to spectra. Let $\mathtt S$ denote the sphere spectrum. There is a sequence of spectra
$$
\mathtt S=Sp^1(\mathtt S)\to\cdots \to Sp^n(\mathtt S)\to \cdots Sp^\infty(\mathtt S)=H\mathbb Z
$$
Note that $Sp^\infty(\mathtt S)=H\mathbb Z$ by the Dold-Thom theorem. Let us call this sequence the symmetric powers filtration of the Eilenberg Mac Lane spectrum.


The point of this is that it is easy to see that the spectra $\partial\_1 F\_n$ are equivalent to the *subquotients* of the symmetric powers filtration.


$$
\partial\_1 F\_n\simeq Sp^n(\mathtt S)/Sp^{n-1}(\mathtt S)
$$


**Remark** The equivalence $Sp^n(\mathtt S)/Sp^{n-1}(\mathtt S)\simeq \Sigma^\infty \Sigma B\_{\mathrm{nt}} \Sigma\_n$ was initially discovered by Kathryn Lesh.


The spectra $Sp^n(\mathtt S)/Sp^{n-1}(\mathtt S)$ have been studied quite a lot, from various perspectives. The following result summarizes some facts about them that were known for a long time.


**Theorem** The spectrum $Sp^n(\mathtt S)/Sp^{n-1}(\mathtt S)$ and therefore also $\partial\_1 F\_n$ has the following properties


1. It is rationally trivial for $n>1$
2. It is integrally trivial unless $n$ is a power of a prime.
3. If $n$ is a power of a prime $p$ then it is a $p$-local spectrum.


When $n=p^k$, the mod $p$-homology of $Sp^{p^k}(\mathtt S)/Sp^{p^k-1}(\mathtt S)$ is well-understood. It was initially calculated by Nakaoka. You can learn more cool details about these spectra for example in [these](https://www.math.ias.edu/%7Elurie/ThursdayFall2017/Lecture13-Symmetric-power.pdf) notes of a lecture by Mike Hopkins.




---


Now let's consider $\partial\_k F\_n$ for $k\ge 1$. You can express $\partial\_k F\_n$ in terms of $\partial\_1 F\_m$ (for varying $m$) using Goodwillie's model for $\partial\_k$ as the multi-linearized cross-effect. The $k$-th cross effect of $F\_n$ is the following functor of $k$ variables
$$
\bigvee\_{n\_1+\cdots + n\_k=n, n\_i>0} F\_{n\_1}(X\_1)\wedge \ldots \wedge F\_{n\_k}(X\_k) 
$$
After multi-linearization, we obtain the following formula
$$
\partial\_k F\_n\simeq \bigvee\_{n\_1+\cdots + n\_k=n, n\_i>0} \partial\_1 F\_{n\_1}\wedge \ldots \wedge \partial\_1 F\_{n\_k}
$$
where the action of $\Sigma\_k$ is self-evident. For example
$$
\partial\_2 F\_4 \simeq \partial\_1 F\_3 \wedge \partial\_1 F\_1 \vee \partial\_1 F\_1 \wedge \partial\_1 F\_3 \vee \partial\_1 F\_2\wedge \partial\_1 F\_2
$$
Where $\Sigma\_2$ acts by permuting the first two wedge factors, and also permuting the two smash factors of the last wedge factor.


