arXiv subjects

Shangjie Zhang

Publications and source records attributed to Shangjie Zhang.

6 recordsLinked to original sources

arXiv2026

$C_{p^n}$-equivariant Mahowald invariants

The classical Mahowald invariant is an operation that systematically produces new elements in the stable homotopy groups of spheres from known ones. We introduce the $C_{p^n}$-Mahowald invariant: a relation $π_\star S_{C_{p^{n-1}}} \rightharpoonup π_\ast S$ between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when $n=1$. We compute the $C_{p^n}$-Mahowald invariants of all elements in the Burnside ring $A(C_{p^{n-1}}) = π_0 S_{C_{p^{n-1}}}$, extending Mahowald and Ravenel's computation of $M_{C_p}(p^k)$. As a consequence, we determine the image of the $C_p$-geometric fixed point map $Φ^{C_p} : π_V S_{C_{p^n}} \to π_0 S_{C_{p^n}/C_p} \cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$.

math.AT↗

arXiv2026

A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres

We determine the $RO(C_2)$-graded Hurewicz images of the $C_2$-equivariant Eilenberg--MacLane spectra $H\underline{\mathbb F_2}$, $H\underline{\mathbb Z}$ and $H\underline{A}$, where $\underline{\mathbb F_2}$ and $\underline{\mathbb Z}$ denote the constant Mackey functors with values in $\mathbb F_2$ and $\mathbb Z$, respectively, and $\underline A$ denotes the Burnside Mackey functor. Surprisingly, the answer is closely tied to the problem of vector fields on spheres: the element $\fracθ{ρ^kτ^n}$ in the negative cone of the homotopy groups of $H\underline{\mathbb F_2}$ lies in the Hurewicz image if and only if $S^n$ admits $k$ linearly independent vector fields. Moreover, using the Generalized Leibniz Rule and the Generalized Mahowald Trick introduced by arXiv:2412.10879, we show that there are nonzero Adams differentials of arbitrary length supported by filtration-$0$ elements in the genuine $C_2$-equivariant Adams spectral sequence.

math.AT↗

arXiv2026

$C_3$-equivariant stable stems

We compute the spoke-graded $C_3$-equivariant stable homotopy groups of spheres $π_{i, j}^{C_3}$, for stems less than 25 (i.e. $i\leq 25$) and for weights between -16 and 16 (i.e. $-16\leq j\leq 16$). In particular, for $j=2k$, this corresponds to the usual $RO(C_3)$-graded homotopy groups of spheres $π^{C_3}_{i-j+kλ}$ for some fixed 2-dimensional $C_3$-faithful representation $λ$. We also describe the geometric fixed point map $Φ^{C_3}: π_{i, j}^{C_3}\to π_{i-j}^{cl}$ and the underlying map $Res: π_{i, j}^{C_3}\to π_{i}^{cl}$.

math.AT↗

arXiv2025

Nonresonant Raman control of ferroelectric polarization

Important advances have recently been made in the search for materials with complex multi-phase landscapes that host photoinduced metastable collective states with exotic functionalities. In almost all cases so far, the desired phases are accessed by exploiting light-matter interactions via the imaginary part of the dielectric function through above-bandgap or resonant mode excitation. Nonresonant Raman excitation of coherent modes has been experimentally observed and proposed for dynamic material control, but the resulting atomic excursion has been limited to perturbative levels. Here, this challenge is overcome by employing nonresonant ultrashort pulses with low photon energies well below the bandgap. Using mid-infrared pulses, ferroelectric reversal is induced in lithium niobate, and the large-amplitude mode displacements are characterized through femtosecond stimulated Raman scattering and second harmonic generation. This approach, validated by first-principle calculations, defines a novel method for synthesizing hidden phases with unique functional properties and manipulating complex energy landscapes at reduced energy consumption and ultrafast speeds.

physics.optics↗

arXiv2025

Equivariant $v_{1,\vec{0}}$-self maps

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in π_0 S_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a self map $v\colonΣ^V C(X)_{(p)}\rightarrow C(X)_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These are transchromatic $v_{1,\vec{0}}$-self maps, in the sense that they are lifts of classical $v_1$-self maps for which the telescope $C(X)_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points.

math.AT↗

arXiv2022

The reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems

We describe in terms of generators and relations the ring structure of the $RO(C_2)$-graded $C_2$-equivariant stable stems $π_\star^{C_2}$ modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational $C_2$-equivariant sphere $π_\star^{C_2}(S_\mathbb{Q})$.

math.AT↗