Real motivic and $C_2$-equivariant Mahowald invariants

DOI10.1112/TOPO.12185arXiv1904.12996MaRDI QIDQ6317970

Publication date: 29 April 2019

Abstract: We generalize the Mahowald invariant to the

m a t h b b R

-motivic and

C_{2}

-equivariant settings. For all

i > 0

with

i e q u i v 2, 3 m o d 4

, we show that the

m a t h b b R

-motivic Mahowald invariant of

(2 + h o e t a)^{i} i n p i_{0, 0}^{m a t h b b R} (S^{0, 0})

contains a lift of a certain element in Adams' classical

v_{1}

-periodic families, and for all

i > 0

, we show that the

m a t h b b R

-motivic Mahowald invariant of

e t a^{i} i n p i_{i, i}^{m a t h b b R} (S^{0, 0})

contains a lift of a certain element in Andrews'

m a t h b b C

-motivic

w_{1}

-periodic families. We prove analogous results about the

C_{2}

-equivariant Mahowald invariants of

(2 + h o e t a)^{i} i n p i_{0, 0}^{C_{2}} (S^{0, 0})

and

e t a^{i} i n p i_{i, i}^{C_{2}} (S^{0, 0})

by leveraging connections between the classical, motivic, and equivariant stable homotopy categories. The infinite families we construct are some of the first periodic families of their kind studied in the

m a t h b b R

-motivic and

C_{2}

-equivariant settings.

Mathematics Subject Classification ID

Equivariant homotopy theory in algebraic topology (55P91) Equivariant homotopy groups (55Q91) Stable homotopy of spheres (55Q45) (v_n)-periodicity (55Q51)

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