Root invariants in the Adams spectral sequence

Behrens, Mark

doi:10.1090/S0002-9947-05-03773-6

Root invariants in the Adams spectral sequence
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Trans. Amer. Math. Soc. 358 (2006), 4279-4341 Request permission

Abstract:

Let $E$ be a ring spectrum for which the $E$-Adams spectral sequence converges. We define a variant of Mahowald’s root invariant called the ‘filtered root invariant’ which takes values in the $E_1$ term of the $E$-Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the $E$-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime $2$. We use the filtered root invariants to compute some low-dimensional root invariants of $v_1$-periodic elements at the prime $3$. We also compute the root invariants of some infinite $v_1$-periodic families of elements at the prime $3$.

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