Gregory Arone

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

Filtered spectra arising from permutative categories

\mathbb{R}^m

Calculus of functors, operad formality, and rational homology of embedding spaces

into

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

\mathbb{R}^n

Loop structures in Taylor towers

. We view the space of embeddings as the value of a certain functor at

Bredon homology of partition complexes

\mathbb{R}^m

Coformality and rational homotopy groups of spaces of long knots

, 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

Augmented Gamma-spaces, the stable rank filtration, and a bu analogue of the Whitehead conjecture

2m+1<n

Derivatives of embedding functors I: the stable case

, 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.

The Action of Young Subgroups on the Partition Complex

Abstract Given a special Γ-category 𝒞 satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to 𝒞 and the Eilenberg-Mac Lane spectrum Hℤ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of Hℤ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, {Am }, that interpolate between bu and . We think of Am as a “bu-analogue” of Sp m (S) and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the mth subquotient is contractible unless m is a power of a prime, and in vk -periodic homotopy the filtration has only k + 2 non-trivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weisss orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {Am }, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of bu. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture.