# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

Lemma 3.2.3.5. Fix $1 \leq i \leq n$. Then there is a unique function $m_ i: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x)$ with the following property:

$(\ast )$

Let $\eta _{i-1}$, $\eta _{i}$, and $\eta _{i+1}$ be elements of $\pi _{n}(X,x)$. Then the $(n+2)$-tuple

$( [e], \ldots , [e], \eta _{i-1}, \eta _{i}, \eta _{i+1}, [e], \ldots , [e] )$

bounds if and only if $\eta _{i} = m_ i( \eta _{i-1}, \eta _{i+1} )$.