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Eva Belmont

Publications and source records attributed to Eva Belmont.

16 recordsLinked to original sources
arXiv2025

A deformation of Borel equivariant homotopy

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.

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arXiv2025

Towards the $p=3$ Kervaire Invariant Problem: The $E_2$-page for the homotopy fixed points spectral sequence computing $π_*({E_6}^{hC_9})$

Hill, Hopkins, and Ravenel suggest that the last remaining Kervaire invariant problem, the case of $p=3$, can be solved by computing the homotopy fixed points spectral sequence for $π_* E_6^{hC_9}$. We prove a detection theorem for this case and use a conjectural form of the $C_9$-action on $E_6$ to compute the $E_2$ page of this spectral sequence away from homological degree zero.

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arXiv2024

A Toda bracket convergence theorem for multiplicative spectral sequences

Moss' theorem, which relates Massey products in the $E_r$-page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable topological model category, we prove a general version of Moss' theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. The theorem has broad applications, e.g. to the computation of the motivic slice and motivic Adams spectral sequences.

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arXiv2023

Bredon homological stability for configuration spaces of $G$-manifolds

McDuff and Segal proved that unordered configuration spaces of open manifolds satisfy homological stability: there is a stabilization map $σ: C_n(M)\to C_{n+1}(M)$ which is an isomorphism on $H_d(-;\mathbb{Z})$ for $n\gg d$. For a finite group $G$ and an open $G$-manifold $M$, under some hypotheses we define a family of equivariant stabilization maps $σ_{G/H}:C_n(M)\to C_{n+|G/H|}(M)$ for $H\leq G$. In general, these do not induce stability for Bredon homology, the equivariant analogue of singular homology. Instead, we show that each $σ_{G/H}$ induces isomorphisms on the ordinary homology of the fixed points of $C_n(M)$, and if the group is Dedekind (e.g. abelian), we obtain the following Bredon homological stability statement: $H^G_d(\bigsqcup_{n\geq 0}C_n(M))$ is finitely generated over $\mathbb{Z}[σ_{G/H} : H\leq G]$. This reduces to the classical statement when $G=e$.

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arXiv2023

Normalizer decompositions of p-local compact groups

We give a normalizer decomposition for a p-local compact group (S, F, L) that describes |L| as a homotopy colimit indexed over a finite poset. Our work generalizes the normalizer decompositions for finite groups due to Dwyer, for p-local finite groups due to Libman, and for compact Lie groups in separate work due to Libman. Our approach gives a result in the Lie group case that avoids topological subtleties with Quillen's Theorem A, because we work with discrete groups. We compute the normalizer decomposition for the p-completed classifying spaces of U(p) and SU(p) and for the p-compact groups of Aguade and Zabrodsky.

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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 $π_\star^{C_2}(S_\mathbb{Q})$.

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arXiv2022

Beta families arising from a $v_2^9$ self map on $S/(3,v_1^8)$

We show that $v_2^9$ is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing $π_*(S/(3,v_1^8))$, and use this to conclude that the families $β_{9t+3/i}$ for $i=1,2$, $β_{9t+6/i}$ for $i=1,2,3$, $β_{9t+9/i}$ for $i=1,\dots,8$, $α_1β_{9t+3/3}$, and $α_1β_{9t+7}$ are permanent cycles in the 3-primary Adams-Novikov spectral sequence for the sphere for all $t\geq 0$. We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams-Novikov $E_2$ page for the sphere in relevant degrees. The $i=1$ cases recover previously known results of Behrens-Pemmaraju and the second author. The results about $β_{9t+3/3}$, $β_{9t+6/3}$ and $β_{9t+8/9}$ were previously claimed by the second author; the computer calculations allow us to give a more direct proof. As an application, we determine the image of the Hurewicz map $π_*S \to π_*tmf$ at $p=3$.

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arXiv2022

$\mathbb{R}$-motivic $v_1$-periodic homotopy

We compute the $v_1$-periodic $\mathbb{R}$-motivic stable homotopy groups. The main tool is the effective slice spectral sequence. Along the way, we also analyze $\mathbb{C}$-motivic and $η$-periodic $v_1$-periodic homotopy from the same perspective.

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arXiv2021

Normalizers of chains of discrete $p$-toral subgroups in compact Lie groups

In this paper we study the normalizer decomposition of a compact Lie group $G$ using the information of the fusion system $\mathcal{F}$ of $G$ on a maximal discrete $p$-toral subgroup. We prove that there is an injective map from the set of conjugacy classes of chains of $\mathcal{F}$-centric, $\mathcal{F}$-radical discrete $p$-toral subgroups to the set of conjugacy classes of chains of $p$-centric, $p$-stubborn continuous $p$-toral subgroups. The map is a bijection when $π_0(G)$ is a finite $p$-group. We also prove that the classifying space of the normalizer of a chain of discrete $p$-toral subgroups of $G$ is mod $p$ equivalent to the classifying space of the normalizer of the corresponding chain of $p$-toral subgroups.

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arXiv2020

A new approach to mod 2 decompositions of BSU(2) and BSO(3)

Dwyer, Miller and Wilkerson proved that at the prime 2, the classifying spaces of SU(2) and SO(3) can be obtained as a homotopy pushout of the classifying spaces of certain subgroups. In this paper we show explicitly how these decompositions arise from the fusion systems of SU(2) and SO(3) over maximal discrete 2-toral subgroups.

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arXiv2020

R-motivic stable stems

We compute some R-motivic stable homotopy groups. For $s - w \leq 11$, we describe the motivic stable homotopy groups $π_{s,w}$ of a completion of the R-motivic sphere spectrum. We apply the $ρ$-Bockstein spectral sequence to obtain R-motivic Ext groups from the C-motivic Ext groups, which are well-understood in a large range. These Ext groups are the input to the R-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by $ρ$, 2, and $η$. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.

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arXiv2019

A Cartan-Eilenberg spectral sequence for a non-normal extension

Let $Φ\to Γ\to Σ$ be a conormal extension of Hopf algebras over a commutative ring $k$, and let $M$ be a $Γ$-comodule. The Cartan-Eilenberg spectral sequence $$ E_2 = \mathrm{Ext}_Φ(k,\mathrm{Ext}_Σ(k,M)) \implies \mathrm{Ext}_Γ(k,M)$$ is a standard tool for computing the Hopf algebra cohomology of $Γ$ with coefficients in $M$ in terms of the cohomology of the pieces $Φ$ and $Σ$. Bruner and Rognes, generalizing a construction of Davis and Mahowald, have introduced a generalization of the Cartan-Eilenberg spectral sequence converging to $\mathrm{Ext}_Γ(k,M)$ that can be defined when $Φ= Γ\square_Σk$ is compatibly an algebra and a $Γ$-comodule. We offer a concrete cobar-like construction that fits into their framework, and show how this work fits into a larger story. In particular, we show that this spectral sequence is isomorphic, starting at the $E_1$ page, to both the Adams spectral sequence in the stable category of $Γ$-comodules as studied by Margolis and Palmieri, and to a filtration spectral sequence on the cobar complex for $Γ$ originally due to Adams. We obtain a description of the $E_2$ term under an additional flatness assumption. We discuss applications to computing localizations of the Adams spectral sequence $E_2$ page.

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arXiv2019

Localizing the $E_2$ page of the Adams spectral sequence

There is only one nontrivial localization of $π_*S_{(p)}$ (the chromatic localization at $v_0=p$), but there are infinitely many nontrivial localizations of the Adams $E_2$ page for the sphere. The first non-nilpotent element in the $E_2$ page after $v_0$ is $b_{10}\in \mathrm{Ext}_A^{2p(p-1)-2}(\mathbb{F}_p,\mathbb{F}_p)$. We work at $p=3$ and study $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ (where $P$ is the algebra of dual reduced powers), which agrees with the infinite summand $\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ of $\mathrm{Ext}_A(\mathbb{F}_3,\mathbb{F}_3)$ above a line of slope ${1\over 23}$. We compute up to the $E_9$ page of an Adams spectral sequence in the category $\mathrm{Stable}(P)$ converging to $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$, and conjecture that the spectral sequence collapses at $E_9$. We also give a complete calculation of $b_{10}^{-1}\mathrm{Ext}_P^*(\mathbb{F}_3,\mathbb{F}_3[ξ_1^3])$.

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arXiv2015

l-adic properties of partition functions

Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ to establish remarkable ``self-similarity'' properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers $p_r$ of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for $p_r$ and spt on arithmetic progressions of the form $\ell^mn+δ$ modulo powers of $\ell$. Our work gives a conceptual explanation of the exceptional congruences of $p_r$ observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

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