arXiv subjects

Gregory Arone

Publications and source records attributed to Gregory Arone.

At least 19 recordsLinked to original sources

arXiv2025

Polynomial functors from free groups to a stable infinity-category

We study the category of polynomial functors from finitely generated free groups to a stable infinity-category D. We show that this category is equivalent to the category of excisive functors from pointed animas to D, and also to truncated right comodules over the commutative operad with values in D. The latter formulation generalizes a result of Geoffrey Powell in characteristic zero. We use the equivalence of categories to calculate Ext between polynomial functors from free groups to abelian groups, extending previous results of Christine Vespa and others. Using the work of Aurelien Djament, we give applications to stable cohomology of automorphism groups of free groups with coefficients in a polynomial functor.

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arXiv2024

The spectrum of excisive functors

We prove a thick subcategory theorem for the category of $d$-excisive functors from finite spectra to spectra. This generalizes the Hopkins-Smith thick subcategory theorem (the $d=1$ case) and the $C_2$-equivariant thick subcategory theorem (the $d=2$ case). We obtain our classification theorem by completely computing the Balmer spectrum of compact $d$-excisive functors. A key ingredient is a non-abelian blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. Also important are the techniques of tensor triangular geometry and striking analogies between functor calculus and equivariant homotopy theory. In particular, we introduce a functor calculus analogue of the Burnside ring and describe its Zariski spectrum \`{a} la Dress. The analogy with equivariant homotopy theory is strengthened further through two applications: We explain the effect of changing coefficients from spectra to ${\mathrm{H}\mathbb{Z}}$-modules and we establish a functor calculus analogue of transchromatic Smith-Floyd theory as developed by Kuhn-Lloyd. Our work offers a new perspective on functor calculus which builds upon the previous approaches of Arone-Ching and Glasman.

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arXiv2023

Intrinsic convergence of the homological Taylor tower for $r$-immersions in $\mathbb R^n$

For an integer $r\ge 2$, the space of $r$-immersions of $M$ in $\R^n$ is defined to be the space of immersions of $M$ in $\R^n$ such that at most $r-1$ points of $M$ are mapped to the same point in $\R^n$. The space of $r$-immersions lies ``between" the embeddings and the immersions. We calculate the connectivity of the layers in the homological Taylor tower for the space of $r$-immersions in $\mathbb R^n$ (modulo immersions), and give conditions that guarantee that the connectivity of the maps in the tower approaches infinity as one goes up the tower. We also compare the homological tower with the homotopical tower, and show that up to degree $2r-1$ there is a ``Hurewicz isomorphism" between the first non-trivial homotopy groups of the layers of the two towers.

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arXiv2022

The space of $r$-immersions of a union of discs in $\mathbb R^n$

For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb R^n$ is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that the preimage of every point in $\mathbb R^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb R^n$ and the $k^{\text{th}}$ power of the space of injective linear maps from $\mathbb R^m$ to $\mathbb R^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is ``well-known'', but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions.

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arXiv2021

The Action of Young Subgroups on the Partition Complex

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|Π_n|$, which is the $Σ_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|Π_n|^G$ in terms of subgroup posets for general $G\subset Σ_n$ and prove a formula for the restriction of $|Π_n|$ to Young subgroups $Σ_{n_1}\times \dots\times Σ_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover surprising links between strict Young quotients of $|Π_n|$, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|Π_n|^\diamond\wedge_{Σ_n} (S^\ell)^{\wedge n}$ and give a combinatorial proof of a splitting in derived algebraic geometry. Combining all our results, we decompose strict Young quotients of $|Π_n|$ in terms of "atoms" $|Π_d|^\diamond\wedge_{Σ_d} (S^\ell)^{\wedge d}$ for $\ell$ odd and compute their homology. We thereby also generalise Goerss' computation of the algebraic André-Quillen homology of trivial square-zero extensions from $\mathbb{F}_2$ to $\mathbb{F}_p$ for $p$ an odd prime.

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arXiv2021

Noncommutative CW-spectra as enriched presheaves on matrix algebras

Motivated by the philosophy that $C^*$-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of $C^*$-algebras. We focus on $C^*$-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable $\infty$-category of noncommutative CW-spectra, which we denote by $\mathtt{NSp}$. Let $\mathcal{M}$ be the full spectral subcategory of $\mathtt{NSp}$ spanned by "noncommutative suspension spectra" of matrix algebras. Our main result is that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. To prove this we first prove a general result which states that any compactly generated stable $\infty$-category is naturally equivalent to the $\infty$-category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an $\infty$-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched $\infty$-categories as developed by Hinich [Hin2,Hin3]. We end by presenting a "strict" model for $\mathcal{M}$. That is, we define a category $\mathcal{M}_s$ strictly enriched in a certain monoidal model category of spectra $\mathtt{Sp^M}$. We give a direct proof that the category of $\mathtt{Sp^M}$-enriched presheaves $\mathcal{M}_s^{op}\to\mathtt{Sp^M}$ with the projective model structure models $\mathtt{NSp}$ and conclude that $\mathcal{M}_s$ is a strict model for $\mathcal{M}$.

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arXiv2021

Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

In a companion paper [ABS1] we introduced the stable $\infty$-category of noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. Let $\mathcal{M}$ denote the full spectrally enriched subcategory of $\mathtt{NSp}$ whose objects are the non-commutative suspension spectra of matrix algebras. In [ABS1] we proved that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. In this paper we investigate the structure of $\mathcal{M}$, and derive some consequences regarding the structure of $\mathtt{NSp}$. To begin with, we introduce a rank filtration of $\mathcal{M}$. We show that the mapping spectra of $\mathcal{M}$ map naturally to the connective $K$-theory spectrum $ku$, and that the rank filtration of $\mathcal{M}$ is a lift of the classical rank filtration of $ku$. We describe the subquotients of the rank filtration in terms of complexes of direct-sum decompositions which also arose in the study of $K$-theory and of Weiss's orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of $\mathtt{NSp}$ as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the $p$-localization and the chromatic localization of $\mathcal{M}$.

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arXiv2021

Embedding obstructions in ${\mathbb R}^d$ from the Goodwillie-Weiss calculus and Whitney disks

Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embedding $K$ into a Euclidean space $\mathbb{R}^d$. For $2$-dimensional complexes in $\mathbb{R}^4$, a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner. We focus on the first obstruction beyond the classical embedding obstruction of van Kampen. In this case we show the two approaches lead to essentially the same obstruction. We also give another geometric interpretation of our obstruction, as a triple collinearity condition. Furthermore, we relate our obstruction to the Arnold class in the cohomology of configuration spaces. The obstructions are shown to be realized in a family of examples. Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometric and cohomological theories.

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arXiv2019

Spaces of knotted circles and exotic smooth structures

Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension $n$ that are homeomorphic. We prove that the spaces of smooth knots $Emb(S^1, N_1)$ and $Emb(S^1, N_2)$ have the same homotopy $(2n-7)$-type. In the 4-dimensional case this means that the spaces of smooth knots in homeomorphic 4-manifolds have sets $\pi_0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi_1$. The result about $\pi_0$ is well-known and elementary, but the result about $\pi_1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie-Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie-Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in $N$ does not depend on the smooth structure on $N$. Our results also give a lower bound on $\pi_2 Emb(S^1, N)$. We use our model to show that for every choice of basepoint, each of the homotopy groups $\pi_1$ and $\pi_2$ of $Emb(S^1, S^1\times S^3)$ contains an infinitely generated free abelian group.

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arXiv2019

Hyperbolic Nodal Band Structures and Knot Invariants

We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum- and lattice models with relatively short-ranged hopping that is within experimental reach. To determine the topology of the nodal structures, we devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.

cond-mat.mes-hall↗

arXiv2019

Goodwillie Calculus

We survey the theory and applications of Goodwillie's calculus of homotopy functors and related topics.

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arXiv2018

Fixed points of coisotropic subgroups of $Γ_{k}$ on decomposition spaces

We study the equivariant homotopy type of the poset of orthogonal decompositions of a finite-dimensional complex vector space. Suppose that n is a power of a prime p, and that D is an elementary abelian p-subgroup of U(n) acting on complex n-space by the regular representation. We prove that the fixed point space of D acting on the decomposition poset of complex n-space contains as a retract the unreduced suspension of the Tits building for GL(k), which a wedge of (k-1)-dimensional spheres. Let Gamma be the projective elementary abelian subgroup of U(n) that contains the center of U(n) and acts irreducibly on complex n-space. We prove that the fixed point space of Gamma acting on the space of proper orthogonal decompositions of complex n-space is homeomorphic to a symplectic Tits building, which is also a wedge of (k-1)-dimensional spheres. As a consequence of these results, we find that the fixed point space of any coisotropic subgroup of Gamma contains, as a retract, a wedge of (k-1)-dimensional spheres. We make a conjecture about the full homotopy type of the fixed point space of D, based on a more general branching conjecture, and we show that the conjecture is consistent with our results.

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arXiv2015

A branching rule for partition complexes

Let $S_n$ be the symmetric group, and let $Y$ be a Young subgroup of $S_n$. Let $Π_n$ be the complex of partitions of $\{1, \ldots, n\}$. Our main result is a $Y$-equivariant decomposition of $Π_n$. As an application, we obtain new information about the quotient space of $|Π_n|$ by a Young subgroup.

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arXiv2014

A classification of Taylor towers of functors of spaces and spectra

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad. For functors with values in the category of spectra, we give a rather explicit description of the associated comonads and their coalgebras. In particular, for functors from based spaces to spectra we interpret this new structure as that of a divided power right module over the operad formed by the derivatives of the identity on based spaces.

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arXiv2014

Manifolds, K-theory and the calculus of functors

The Taylor tower of a functor from based spaces to spectra can be classified according to the action of a certain comonad on the collection of derivatives of the functor. We describe various equivalent conditions under which this action can be lifted to the structure of a module over the Koszul dual of the little L-discs operad. In particular, we show that this is the case when the functor is a left Kan extension from a certain category of `pointed framed L-manifolds' and pointed framed embeddings. As an application we prove that the Taylor tower of Waldhausen's algebraic K-theory of spaces functor is classified by an action of the Koszul dual of the little 3-discs operad.

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arXiv2014

Cross-effects and the classification of Taylor towers

Let F be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of F have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of F are a right module over a certain `pro-operad'. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor F can be reconstructed from this structure on the derivatives.

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arXiv2013

On the rational homology of high dimensional analogues of spaces of long knots

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into $\mathbb{R}^n$. We view the space of embeddings as the value of a certain functor at $\mathbb{R}^m$, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when $2m+1<n$, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.

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arXiv2013

Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on different ways how the calculations can be done. In particular we describe three different graph-complexes computing the rational homotopy of spaces of long embeddings. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

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