Remark 3.2.3.3. Let $\vec{\eta } = (\eta _0, \eta _1, \ldots , \eta _{n+1} )$ be an $(n+2)$-tuple of elements of $\pi _{n}(X,x)$, so that we can write $\eta _ i = [ \sigma _ i ]$ for some $n$-simplex $\sigma _{i} \in \Sigma $. We will say that the tuple of homotopy classes $\vec{\eta }$ bounds if the tuple of simplices $\vec{\sigma } = ( \sigma _0, \sigma _1, \ldots , \sigma _{n+1} )$ bounds, in the sense of Notation 3.2.3.1. By virtue of Lemma 3.2.3.2, this condition is independent of the choice of $\vec{\sigma }$.
With this terminology, Theorem 3.2.2.10 asserts (in the case $n \geq 2$) that there is a unique abelian group structure on the set $\pi _{n}(X,x)$ with the following pair of properties:
- $(a)$
The identity element of $\pi _{n}(X,x)$ is the homotopy class $[e]$.
- $(b)$
An $(n+2)$-tuple $\vec{\eta } = (\eta _0, \eta _1, \ldots , \eta _{ n+1} )$ bounds if and only if the sum $\sum _{i=0}^{n+1} (-1)^{i} \eta _ i$ vanishes in $\pi _{n}(X,x)$.