# How to combine shuffles to prove associativity of Eilenberg-Zilber map

I've got a problem related to $$(p,q)$$-shuffles that comes from the Eilenberg-Zilber map $$\nabla$$ when I tried to show that this map is associative in the sense that $$\nabla(\nabla\otimes 1)=\nabla(1\otimes \nabla)$$.

On one side, I have for $$(p,q)$$-shuffles $$(\mu,\nu)$$ and $$(p+q,r)$$-shuffles $$(\alpha,\beta)$$ the expression

$$\sum_{(\alpha,\beta), (\mu,\nu)} sgn(\alpha,\beta)sgn(\mu,\nu)s_\beta(s_\nu(a))\otimes s_\beta(s_\mu(b))\otimes s_\alpha(c)$$

and on the other side, for $$(q,r)$$-shuffles $$(\delta,\gamma)$$ and $$(p,q+r)$$-shuffles $$(\xi,\omega)$$

$$\sum_{(\xi,\omega),(\delta,\gamma)} sgn(\xi,\omega)sgn(\delta,\gamma)s_\omega(a)\otimes s_\xi(s_\gamma(b))\otimes s_\xi(s_\delta(c))$$

In order to show that the latter expression is the same as the former, I would like to combine $$(\xi,\omega)$$ and $$(\delta,\gamma)$$ to obtain a $$(p,q)$$-shuffle and a $$(p+q,r)$$-shuffle such that the products of its signs is precisely $$sgn(\xi,\omega)sgn(\delta,\gamma)$$.

Sice $$(\xi,\omega)$$ is a $$(p,q+r)$$-shuffle, I can just take $$(\xi_1,\dots, \xi_p,\omega_1,\dots,\omega_q)$$ as $$(p,q)$$-shuffle. But then, to obtain a $$(p+q,r)$$-shuffle I have to permute the $$\xi_i$$ and the $$\omega_j$$ to order them in a way that the sign is the one I want. Somehow I should use $$(\delta,\gamma)$$ to do that, but this is a $$(q,r)$$-shuffle, so I don't know what to do.

• The tags (shuffles) and (chain-complexes) currently don't have any user guidance, which leads to a higher chance they being removed by other users. As the tag creator, would you mind to add some? – YuiTo Cheng Apr 21 at 14:19
• @YuiToCheng I don't have tag wiki edit privileges so I cannot do that. – Javi Apr 21 at 14:29
• Me neither. Still, you can suggest edits. – YuiTo Cheng Apr 21 at 14:30
• @YuiToCheng ok I will – Javi Apr 21 at 14:30