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Erik Lindell

Publications and source records attributed to Erik Lindell.

7 recordsLinked to original sources
arXiv2026

Twisted homological stability for handlebody mapping class groups

We prove twisted homological stability for handlebody mapping class groups. Using the categorical framework developed by Randal-Williams and Wahl, we establish that the homology of the handlebody groups stabilises with respect to both genus and the number of marked boundary discs, for all coefficient systems of finite degree. Our first main theorem refines and extends the twisted stability result for handlebodies outlined by Randal-Williams and Wahl, allowing any number of marked discs and boundary points. We then introduce the notion of coefficient bisystem to treat stability under variation of boundary markings. As an application, we deduce homological stability for moduli spaces of 3-dimensional handlebodies equipped with tangential structures.

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arXiv2025

Stable cohomology of $Aut(F_n)$ with bivariant twisted coefficients

We compute the cohomology groups of the automorphism group of the free group $F_n$, with coefficients in arbitrary tensor products of the standard representation $H_1(F_n, \mathbb{Q})$ and its dual, in a range where $n$ is sufficiently large compared to the cohomological degree and the number of tensor factors.

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arXiv2025

Plethysm for characters of relative operads and PROPs

We investigate the relationship between symmetric functions and the representation theory of operads, relative operads, and props. We extend the classical character map for symmetric sequences to relative bisymmetric sequences and symmetric bimodules. We introduce new operations on symmetric functions, the relative plethysm and the (connected) box product, which model via the character map the composition product of relative operads and the box product of prop(erad)s. As applications, we include the computation of characters for stable twisted cohomology of automorphism groups of free groups and the Albanese cohomology of the IA-automorphism group.

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arXiv2024

The walled Brauer category and stable cohomology of $\mathrm{IA}_n$

The IA-automorphism group is the group of automorphisms of the free group $F_n$ that act trivially on the abelianization $F_n^{\mathrm{ab}}$. This group is in many ways analoguous to Torelli groups of surfaces and their higher dimensional analogues. In recent work, the stable rational cohomology of such groups was studied by Kupers and Randal-Williams, using the machinery of so-called Brauer categories. In this paper, we adapt their methods to study the stable rational cohomology of the IA-automorphism group. We obtain a conjectural description of the algebraic part of the stable rational cohomology and prove that it holds up to degree $Q+1$, given the assumption that the stable cohomology groups are stably finite dimensional in degrees up to $Q$. In particular, this allows us to compute the algebraic part of the stable cohomology in degree 2, which we show agrees with the part generated by the first cohomology group via the cup product map and which has previously been computed by Pettet. In the appendix, written by Mai Katada, it is shown how the results of the paper can be applied to compute the stable Albanese (co)homology of the IA-automorphism group.

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arXiv2023

Representation stability for homotopy automorphisms

We consider in parallel pointed homotopy automorphisms of iterated wedge sums of topological spaces and boundary relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces and manifolds, we prove that the rational homotopy groups of these homotopy automorphisms form finitely generated FI-modules, and thus satisfy representation stability for symmetric groups, in the sense of Church and Farb.

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arXiv2020

Abelian Cycles in the Homology of the Torelli group

In the early 1980's, Johnson defined a homomorphism $\mathcal{I}_{g}^1\to\bigwedge^3 H_1(S_{g},\mathbb{Z})$, where $\mathcal{I}_{g}^1$ is the Torelli group of a closed, connected and oriented surface of genus $g$ with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding pair maps, in order to compute a large quotient of $H_n(\mathcal{I}_{g}^1,\mathbb{Q})$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal{I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n(\mathcal{I}_{g,1})$ for $n\ge 2$ and $g$ large enough.

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