Orthogonal Calculus Research Articles

Comparing orthogonal calculus and calculus with Reality

We show that there exists a suitable C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus “up to a shift” in an analogous way with the recovery of real topological K-theory from Atiyah’s K-theory with Reality via appropriate C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}-fixed points.

The localization of orthogonal calculus with respect to homology

The localization of orthogonal calculus with respect to homology

Recovering unitary calculus from calculus with reality

By analogy with complex K–theory and K–theory with reality, there are theories of unitary functor calculus and unitary functor calculus with reality, both of which are generalisations of Weiss' orthogonal calculus. In this paper we show that unitary functor calculus can be completely recovered from the unitary functor calculus with reality, in analogy to how complex topological K–theory is completely recovered from K–theory with reality via forgetting the C2–action.

Unitary calculus: model categories and convergence

We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

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

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 {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 Arone et al. (2021) 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 spaces 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}}.

UNITARY FUNCTOR CALCULUS WITH REALITY

AbstractWe construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.

Comparing the orthogonal and unitary functor calculi

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.

Rational orthogonal calculus

We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of O(n). By work of Greenlees and Shipley, we see that these layers are classified by torsion {{mathrm{H}}}^*({{mathrm{B}}}SO(n))[O(n)/SO(n)]-modules.

Comparing the orthogonal and homotopy functor calculi

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor V↦F(SV), where SV is the one-point compactification of V.In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

Capturing Goodwillie's derivative

Recent work of Biedermann and Röndigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with a Σn-action. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.

Model categories for orthogonal calculus

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.

Derivatives of embedding functors I: the stable case

For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula $V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal calculus, in the case when $N$ is a nice open submanifold of a Euclidean space. This leads to a description of the derivatives of this functor when $N$ is a tame stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in $M$, and a certain ``homotopy bundle of spectra'' over this space of trees. The $n$-th derivative is then described as the ``spectrum of restricted sections'' of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor $\Ebar(M, N\times V)$, involving a similar construction with certain spaces of connected graphs (instead of forests) with points marked in $M$.

The block structure spaces of real projective spaces and orthogonal calculus of functors II

For a finite dimensional real vector space V with inner product, let F(V) be the block structure space, in the sense of surgery theory, of the projective space of V. Continuing a program launched in part I, we investigate F as a functor on vector spaces with inner product, relying on functor calculus ideas. It was shown in part I that F agrees with its first Taylor approximation T_1 F (which is a polynomial functor of degree 1) on vector spaces V with dim(V) > 5. To convert this theorem into a functorial homotopy-theoretic description of F(V), one needs to know in addition what T_1 F(V) is when V=0. Here we show that T_1 F(0) is the standard L-theory space associated with the group Z/2, except for a deviation in \pi_0. The main corollary is a functorial two-stage decomposition of F(V) for dim(V) > 5 which has the L-theory of the group Z/2 as one layer, and a form of unreduced homology of RP (V) with coefficients in the L-theory of the trivial group as the other layer. (Except for dimension shifts, these are also the layers in the traditional Sullivan-Wall-Quinn-Ranicki decomposition of F(V). But the dimension shifts are serious and the SWQR decomposition of F(V) is not functorial in V.) Because of the functoriality, our analysis of F(V) remains meaningful and valid when V=R^\infty.

Loop structures in Taylor towers

We study spaces of natural transformations between homogeneous functors in Goodwillie’s calculus of homotopy functors and in Weiss’s orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application uses this description in combination with the Segal Conjecture to obtain a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU.V/. The interest in such deloopings stems from conjectures made by the first and the third author [4] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 55P65; 55P47, 18G55 1 Introduction and notation In this paper, we study homogeneous functors in the sense of Weiss’s orthogonal calculus [13]. More precisely, we calculate the space of natural transformations between such functors, and we give a few examples and applications. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity functor for pointed spaces and in the Weiss tower of the functor V7!BU.V/. The motivation for this delooping theorem will be discussed later in this introduction. To state our results, we summarize the usual notation for this context. Let F be a functor from pointed spaces or from finite-dimensional real or complex vector spaces to pointed spaces or spectra. Goodwillie and Weiss calculus assign to such a functor F a “Taylor” tower of functors ! Pn F! Pn 1 F! ; together with a natural map from F to the homotopy inverse limit of the tower that is often a weak homotopy equivalence. The homotopy fiber of the map Pn F! Pn 1 F is customarily denoted Dn F and is referred to as “the nth homogeneous layer of F ,” while Pn F is referred to as “the nth Taylor polynomial of F .” Depending on whether

Filtered spectra arising from permutative categories

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 Spm(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 Weiss's 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.

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

Let M be a smooth manifold and V a Euclidean space. Let \( \overline{{{\text{Emb}}}} \)(M,V) be the homotopy fiber of the map Emb(M,V) → Imm(M,V). This paper is about the rational homology of \( \overline{{{\text{Emb}}}} \)(M,V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M,V)↦ HQ ∧ \( \overline{{{\text{Emb}}}} \)(M,V)+. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$ (where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$ The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of \( \overline{{{\text{Emb}}}} \)(–,V).

The block structure spaces of real projective spaces and orthogonal calculus of functors

For a finite dimensional real vector space V with inner product, let F(V) be the block structure space, in the sense of surgery theory, of the projective space of V. Continuing a program launchedin (Ma), weinvestigateF asafunctoronvector spaceswith innerproduct,relyingonfunctor calculus ideas. It was shown in (Ma) that F agrees with its first Taylor approximation T1F (which is a polynomial functor of degree 1) on vector spacesV with dim(V) ≥ 6. To convert this theorem into a functorial homotopy-theoretic description of F(V), one needs to know in addition what T1F(V )i s whenV = 0.HereweshowthatT1F(0) isthestandardL-theoryspaceassociatedwith thegroupZ/2, except for a deviation in π0. The main corollary is a functorial two-stage decomposition of F(V )f or dim(V) ≥ 6whichhas theL-theoryofthe groupZ/2asonelayer,andaform ofunreducedhomology ofRP(V) with coefficientsin the L-theory ofthe trivial groupas theother layer.Exceptfor dimension shifts, theseare also the layersin the traditional Sullivan-Wall-Quinn-Ranicki decompositionof F(V). But the dimension shifts are serious and the SWQR decomposition of F(V) is not functorial in V. Becauseof the functoriality, our analysis of F(V) remains meaningful and valid when V = R ∞ .

The Goodwillie Calculus of Functors

The purpose of this meeting was to introduce the participants to calculus of functors, a theory aimed at “approximating” functors in algebra and topology. The focus of the talks was on two related branches of the theory, homotopy and manifold calculus. T he organizers, Thomas Goodwillie and Randy McCarthy, scheduled 16 talks. The first was given by Goodwillie who introduced the main ideas and outlined the plan for the rest of the meeting. In talks 2–5, speakers explained the most important terminology, techniques, and results used in both versions of the theory. In talks 6–8, participants learned about manifold calculus and how it applies to spaces of knots (talk 8), while talks 9–15 dealt with homotopy calculus. In the last four of those lectures, some applications, elaborations on results established in previous talks, and different versions of certain ideas and proofs encountered so far were given. Goodwillie explored some connections between the two versions of calculus of functors (as they apply to the embedding and identity functors) and gave the concluding remarks in the last talk. It should be noted that a third version of the theory, orthogonal calculus, was not discussed due to time constraints, but Goodwillie explained some of its most salient features in an extra evening session. Other evening events took place as well, such as Rainer Vogt's elaboration of an aspect of his talk and informal research reports given by some participants.

The Mitchell-Richter filtration of loops on Stiefel manifolds stably splits

We prove that the Mitchell-Richter filtration of the space of loops on complex Stiefel manifolds stably splits. The result is obtained as a special case of a more general splitting theorem. Another special case is H. Miller’s splitting of Stiefel manifolds. The proof uses the theory of orthogonal calculus developed by M. Weiss. The argument is inspired by an old argument of Goodwillie for a different, but closely related, general splitting result.

Erratum to “Orthogonal calculus”

Notation. ,f is the category of finite dimensional real vector spaces with a positive definite inner product. Morphisms in .1 are the linear maps respecting the inner product. g is the category of continuous functors from .1 to spaces. (The spaces in question are assumed to be compactly generated Hausdorff, homotopy equivalent to CW-spaces). A morphism E > F (natural transformation) in g is an equivalence if E(V) > F(V) is a homotopy equivalence for each V in ,f. An object E in g is polynomial of degree < n if, for each V in .1, the canonical map