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The Smith-Toda complex \(V((p + 1)/2)\) does not exist. (English) Zbl 1194.55017
Let \(p\) be a prime number. A Smith-Toda complex \(V(k)\) is a finite spectrum with Brown-Peterson homology \(\text{BP}_*/(p,v_1,\ldots,v_k)\), where \(v_1,v_2,\ldots\,\) are the standard generators for \(\text{BP}_*\). For \(p\geq 7\) the author shows that there is no Smith-Toda complex \(V\bigl((p+1)/2\bigr)\); also, if there is a complex \(V\bigl((p-1)/2\bigr)\) then it is not a ring spectrum. The proof uses a generalised homotopy fixed point spectral sequence due to Hopkins and Miller.

MSC:
55Q51 \(v_n\)-periodicity
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