Prove: If $q \ge 0$ is an integer and if $z \in Z_n(X)$, then $$\text{cls }z = \text{cls }(Sd^q\space z)$$
What does $Sd^q$ mean? I'm unfamiliar with this notation. This is from Rotman's Algebraic topology on page $116$.
2026-05-15 11:53:44.1778846024
If $Sd : S_*(X) \rightarrow S_*(X)$ is a barycentric subdivision and $q \ge 0$ is an integer, what does $Sd^q$ mean?
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It's just the $q$-fold composition of $Sd$: $Sd^2=Sd\circ Sd$, $Sd^3=Sd\circ Sd^2$ etc.