tmf\mathit{tmf}-based Mahowald invariants

The $2$-primary homotopy $\beta$-family, defined as the collection of Mahowald invariants of Mahowald invariants of $2^i$, $i \geq 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate $\mathit{tmf}$-based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of $\mathit{tmf}$ with trivial $C_2$-action and Behrens' filtered Mahowald invariant machinery.Comment: 52 pages, 41 figures. Comments welcome

Symmetries of exotic spheres via complex and quaternionic Mahowald invariants

We deduce the existence of smooth $U(1)$- and $Sp(1)$-actions on certain exotic spheres using complex and quaternionic analogues of the Mahowald {(root)} invariant. In particular, we prove that the complex (respectively, quaternionic) Mahowald invariant takes an element of $\pi_k^s$ represented by a homotopy sphere $\Sigma^k$ to an element {of $\pi_{k+\ell}$} represented by another homotopy sphere $\Sigma^{k+\ell}$ equipped with a smooth $U(1)$- (respectively, $Sp(1)$-) action with fixed points the original homotopy sphere $\Sigma^k\subset \Sigma^{k+\ell}$. This work is motivated by results of Stolz on the classical Mahowald invariant and smooth $C_2$-actions on homotopy spheres.Comment: 19 pages. Comments welcome

Algebraic slice spectral sequences

For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the $\mathbb{R}$-motivic effective slice spectral sequence is completely determined by the $\rho$-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra. <br

Choosing the Right Consultant

The church in North America is in decline. Research indicates that many churches are not growing or even considered “healthy.” More and more churches are addressing their declining attendance through hiring consultants to identify areas of growth and improvement. The key is finding the right consultant or consulting firm for the needs of the local church. This article seeks to help the local church ask the right questions when selecting a consultant or consulting firm

v1v_1-periodic motivic homotopy over prime fields

We compute the motivic stable homotopy groups of a variant of the connective image-of-$J$ spectrum over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about $v_1$-periodic motivic stable homotopy groups over arbitrary base fields of characteristic not two.Comment: 30 pages, 16 figures. Comments welcome