The Smith-Toda complex \(V((p + 1)/2)\) does not exist
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.
scientific article; zbMATH DE number 5712733
scientific article