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Remark 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 By virtue of Lemma, this condition is independent of the choice of $\vec{\sigma }$.

With this terminology, Theorem 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:


The identity element of $\pi _{n}(X,x)$ is the homotopy class $[e]$.


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 $\pi _{n}(X,x)$.