Gregory Arone
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The spectrum of excisive functors Open
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 th…
Polynomial functors from free groups to a stable infinity-category Open
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…
Intrinsic convergence of the homological Taylor tower for $r$-immersions in $\mathbb R^n$ Open
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 …
The space of $r$-immersions of a union of discs in $\mathbb R^n$ Open
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 $r$ po…
Noncommutative CW-spectra as enriched presheaves on matrix algebras Open
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 noncommutative CW-complexes in the s…
The rank filtration via a filtered bar construction Open
Suppose F is a special Gamma-space equipped with a natural transformation to\nthe infinite symmetric power functor. Segal's infinite loop space machine\nassociates with F a spectrum, denoted kF, equipped with a map to the integral\nEilenbe…
Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra Open
In a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775 , 2021) we introduced the stable $$\infty $$ -category of noncommutative CW-spectra, which we denoted $$\mathtt {NS…
Embedding obstructions in ${\mathbb R}^d$ from the Goodwillie-Weiss calculus and Whitney disks Open
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…
Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra Open
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…
Noncommutative CW-spectra as enriched presheaves on matrix algebras Open
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 t…
Spaces of knotted circles and exotic smooth structures Open
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, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1…
Fixed points of coisotropic subgroups of Γk on decomposition spaces Open
We study the equivariant homotopy type of the poset Lpk of orthogonal decompositions of Cpk. The fixed point space of the p-radical subgroup Γk⊂U(pk) acting on Lpk is shown to be homeomorphic to a symplectic Tits building, a wedge of (k−1)…
Fixed points of coisotropic subgroups of $\Gamma_{k}$ on decomposition spaces Open
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…
Hyperbolic nodal band structures and knot invariants Open
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 dissipati…
Report on 1905.05858v1 Open
We extend the list of known band structure topologies to include hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian r…
Report on 1905.05858v1 Open
We extend the list of known band structure topologies to include hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian r…
Goodwillie Calculus Open
We survey the theory and applications of Goodwillie's calculus of homotopy functors and related topics.
P-Toral Approximations Compute Bredon Homology Open
We study Bredon homology approximations for spaces with an action of a compact Lie group G. We show that if M is a coMackey functor satisfying mild p-locality conditions, then Bredon homology of a G-space X with coefficients in M is determ…
Fixed points of coisotropic subgroups of $Γ_{k}$ on decomposition spaces Open
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…
Manifolds, K-theory and the calculus of functors Open
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 ca…
Cross-effects and the classification of Taylor towers Open
Let [math] be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of [math] have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the …
Bredon homology of partition complexes Open
We prove that the Bredon homology or cohomology of the partition complex with fairly general coefficients is either trivial or computable in terms of constructions with the Steinberg module. The argument involves a theory of Bredon homolog…