 Cup power (source code)

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= Cup power
{title2=$a^r=\underbrace{a\smile\cdots\smile a}_{r\text{ factors}}$}

For a <cohomology class> $a$, its rth cup power is its r-fold product in the <cohomology ring>. Naturality gives $f^*(a^r)=(f^*a)^r$. A nilpotent class has some positive cup power zero; the nilpotence order is preserved by ring isomorphisms and can obstruct a <homotopy equivalence> or a nonzero <mapping degree>.

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