# Dyer–Lashof operations for more than 2 inputs

Let $$\mathcal{O}$$ be a topological operad and $$X$$ an algebra over it. Let the base ring be $$\mathbb{Z}_2$$. If $$C_*$$ denotes the singular chain complex over $$\mathbb{Z}_2$$, the action of $$\mathcal{O}$$ gives us morphisms $$\mu:C_*(\mathcal{O}(r))\otimes_{r} C_*(X)^{\otimes r}\to C_*(X)$$ where $$\otimes_{r}$$ means the tensor product over the group algebra $$\mathbb{Z}_2\mathfrak{S}_r$$ of the $$r$$th symmetric group. Now consider the covering $$\mathcal{O}(r)\to \mathcal{O}(r)/\mathfrak{S}_r$$. For each simplex $$s\in C_*(\mathcal{O}(r)/\mathfrak{S}_r)$$ we choose one of the $$r!$$ lifts $$\widetilde{s}$$ and consider the map $$\Phi:C_*(\mathcal{O}(r)/\mathfrak{S}_r)\times C_*(X)\to C_*(\mathcal{O}(r))\otimes_r C_*(X)^{\otimes r}, (s,a)\mapsto \widetilde{s}\otimes_r a^{\otimes r}.$$ We see a few things immediately:

1. This map is not a map of chain complexes, but linear in the first argument.
2. As we do not have to care about signs, we have $$\tau_*a^{\otimes r}=a^{\otimes r}$$ and the map is independent of the choice of lifts.
3. If $$c\in C_*(\mathcal{O}(r)/\mathfrak{S}_r)$$ and $$a\in C_*(X)$$ are cycles, then $$\Phi(c,a)$$ is a cycle.
4. If $$c\in C_*(\mathcal{O}(r)/\mathfrak{S}_r)$$ is a boundary and $$a\in C_*(X)$$ is a cycle, then $$\Phi(c,a)$$ is a boundary.

Putting all this together, we get a well-defined map $$\Psi:H_*(\mathcal{O}(r)/\mathfrak{S}_r)\times C^{\text{cyc}}_*(X)\to H_*(C_*(\mathcal{O}(r))\otimes_r C_*(X)^{\otimes r})\stackrel{\mu}{\to}H_*(X).$$

Now if we have $$r=2$$ (so we only have a $$2$$-fold covering) we can even show:

1. If $$c\in C_*(\mathcal{O}(r)/\mathfrak{S}_r)$$ is a cycle and $$a+a'\in C_*(X)$$ is a boundary, then $$\Phi(c,a)+\Phi(c,a')$$ is a boundary.

This finally gives us a map $$Q:H_*(\mathcal{O}(2)/\mathfrak{S}_2)\times H_*(X)\to H_*(X)$$ which is used to define Dyer–Lashof operations: If $$\mathcal{O}=\mathcal{C}_{n+1}$$ is the little $$(n+1)$$-cube operad, then $$\mathcal{C}_{n+1}(2)/\mathfrak{S}_2\simeq \mathbb{R}P^n$$ and each $$e_i\in H_i(\mathbb{R}P^n)$$ gives an operation $$Q_i:H_p(X)\to H_{2p+i}(X)$$.

I was wondering if the last ingredient, namely (5), is only true for $$r=2$$. I have not found a counterexample for $$r=3$$ (still, everything over $$\mathbb{Z}_2$$), but I also could not find a proof.

• I think it is easier to look at $r$ a power of $2$, so the next case being $r=4$. I believe that one should be able to get iterated Dyer-Lashof (Kudo-Araki rather since you are working at 2) operations. Apr 17, 2019 at 15:47
• Thank you! So do you think the statement is wrong for $r$ not being a power of $2$? Apr 20, 2019 at 14:46
• Well I only mean that the case $r$ being a power of 2 is the easiest case to investigate. Presumably for other values of $r$, we would get something related to Kudo-Araki operations via the product. Apr 20, 2019 at 18:05