# Functor Calculus and Operads (11w5058)

Arriving in Banff, Alberta Sunday, March 13 and departing Friday March 18, 2011

## Organizers

Michael Ching (University of Georgia)

Nick Kuhn (University of Virginia)

Victor Turchin (Kansas State University)

## Objectives

I. An Introduction to Functor Calculus.

At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.

A simple and familiar example of this is the Euler characteristic $e(X)$, where the local-to-global property for good decompositions takes the form $e(U union V) = e(U) + e(V) - e(U intersect V)$. A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer--Vietoris sequence. Finally one can consider the functor $SP^infty: Top --> Top$, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups $pi_*(SP^infty(X)) = H_*(X)$, the Meyer--Vietoris sequence for homology is thus a consequence of applying $pi_*( )$ to the homotopy pullback square.

It was the insight of Tom Goodwillie in the 1980's that such ``linear'' functors $F: Top --> Top$ form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree n takes appropriate sorts of $(n+1)$--dimensional cubical homotopy pushout diagrams to $(n+1)$--dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

In his analysis, an arbitrary homotopy functor $F: Top --> Top$, maps to a ``Taylor tower'' of fibrations

$... ---> P_nF ---> P_1F- --> P_0F$, where $P_nF$ is the best degree n polynomial approximation to F. The layers of the tower are always infinite loopspaces of a special form: for each d there is a spectrum $d_n(F)$ with an action of the nth symmetric group $S_n$, such that the homotopy fiber of $P_dF(X) --> P_{d-1}F(X)$ is naturally equivalent to the zero space of the homotopy orbits of the spectrum with $S_n$ action $(d_n(F)$ smash $X^n)$. In particular, associated to a functor F is a symmetric sequence in spectra ${d_0(F), d_1(F), d_2(F), ...}$.

Two kinds of developments suggest the centrality of Goodwillie's ideas in modern algebraic and geometric topology.

Firstly, functor calculus comes in a remarkable number of varieties: the features just described persist in the study of functors $F: C --> D$ for many interesting pairs of categories C and D. Functor calculus includes Homotopy Calculus in which C and D are Top or Spectra, Orthogonal Calculus in which C is the category of either real or complex inner product spaces, Andre-Quillen Calculus in which C and D are categories of commutative S--algebras or similar, and Embedding Calculus in which C is the category of open subsets of a fixed smooth manifold M.

Secondly, when one looks at explicit examples of basic functors, lovely and unexpected features have emerged: the symmetric sequences are often modeled by important operads, and the Taylor towers often behave well after either rational localization or localization with respect to a periodic homology theory. Thus the methods of functor calculus should inform two of the currently most popular directions in topology: operad theory, particularly as it relates to geometric topology, and chromatic topology, including elliptic cohomology and variants.

II. The Workshop

The most basic purpose of the workshop will be to extend awareness of what the homotopical methods of functor calculus offer to those who study operad theory, and reciprocally, to examine what the methods of operad theory and chromatic localization theory offer to the analysis of Taylor towers.

The workshop will address structural questions:

* How much additional information is necessary to reconstruct a Taylor tower from knowledge of a functor's associated symmetric sequence of derivatives?

* To what extent does this sort of classification simplify after localization?

* How closely is the study of Taylor towers in very general contexts related to the theory of operads and their modules?

* What is the relationship of calculus to Koszul duality for operads?

The workshop will address computational applications:

* The use of Orthogonal Calculus to further our understanding of how unstable homotopy groups of spheres are built out of stable information.

* The use of Homotopy Calculus and Andre-Quillen calculus to extract new information about the homology of iterated loop spaces.

* The use of Embedding Calculus, coupled with the theory of operads, to obtain detailed formulas for homotopy types, homology groups, and homotopy groups of spaces of embeddings.

We expect participants to include the lead researchers working on such problems as well as postdocs and advanced graduate students looking to begin work in these areas. There seems to be enthusiasm for this workshop: we contacted most of the people listed below about their interest in attending, and everyone of this group answered positively within 24 hours.

III. A Survey of Recent Research

G. Arone and M. Mahowald showed the potential of applying Homotopy Calculus to the one of most formidable problems of algebraic topology: the calculation of the homotopy groups of spheres. Much of what is classically known follows from two themes: relate unstable homotopy to stable homotopy, and study localization with respect to periodic homology theories. Historically, the first of these has been the study of Hopf invariants and related spectral sequences, and the second has focused on localization with respect to K--theory and resulting calculations of $``v_1--periodic''$ homotopy. Arone and Mahowald's analysis of the Taylor tower of the identity functor $I: Top --> Top$, evaluated at $S^m$ for m odd, offers insight into both themes that trumped much work of the previous decades in this area. This was the first example illustrating how, serendipitously, calculus seems to fit beautifully with the methods of stable homotopy theory as chromatically stratified by Morava K--theories and associated $v_n--self$ maps of finite complexes.

Another result showing how towers fit with chromatic information is N. Kuhn's theorem that polynomial endofunctors of Spectra split into their homogeneous pieces after localization with respect to any Morava K--theory. Featured here, and in earlier work of R. McCarthy, is the equivariant Tate construction. Kuhn also has detailed results about the Morava K--theory of infinite loop spaces as a consequence of a highly structured special case of this splitting. Here his arguments make use of what might be termed Andre--Quillen Calculus, as it uses the Taylor tower for the identity functor $I: Alg --> Alg$, where Alg is the category of commutative nonunital S--algebras. The linearization of I is a version of Topological Andre--Quillen homology.

M. Weiss's Orthogonal Calculus is an adaptation of the Taylor tower to functors from the domain category of finite-dimensional real or complex inner product spaces. The idea is to reconstruct such a functor from knowledge of its behavior on "large'' vector spaces. For example, in the real case, the functor sending V to BO(V) gives an elegant tower converging to BO(V), with first term BO. Very recent work of Arone, W. Dwyer, and K. Lesh shows that there are strong ties between orthogonal calculus and homotopy calculus, and uses these ties to obtain otherwise inaccessible delooping results for structure maps in the tower studied by Arone and Mahowald. A next generation of results about the homotopy groups of spheres, both classical and periodic, seems on the horizon.

Working within the Homotopy Calculus setting, the recent paper of Arone and M. Ching signals a new level of sophistication with respect to our understanding of how to reconstruct a Tayor tower from the symmetric sequence that determines its layers. Firstly, the symmetric sequence $d_*(I_C)$ of the identity functor on C will be an operad in the category Sp(C) of symmetric C-spectra. For example, when C is Top, one gets a geometric version of the Lie operad. Secondly, given a functor $F: C --> D$, the symmetric sequence $d_*(F)$ will be a right $d_*(I_C)$--module and a left $d_*(I_D)$--module. There is a comparison of the Tayor tower with a tower constructed solely using these operad module structures, with the difference measured by Tate constuctions. Conjecturally this story will generalize to other situations, e.g. that of Andre-Quillen Calculus, where some investigations are ongoing by M. Mandell and M. Basterra. Furthermore, analysis of the suspension functor $C --> Sp(C)$ and its right adjoint seems to suggest beautiful connections with the theory of Koszul duality of operads as developed by B. Fresse and others.

In a more geometric context, let O(M) be the category of open subsets of a smooth manifold M. Given another manifold N, the space of immersions functor, $Imm( ,N): O(M)^{op} --> Top$, sends pushouts to pullbacks. By constrast, the space of embeddings functor, $Emb( ,N)$, has less favorable local-to-global behavior. However, the Embedding Calculus of Goodwillie and Weiss offers a systematic way to pass from immersions to embeddings. There has been interesting work in this area by Arone, V. Lambrechts, I. Volic, D. Sinha, and V. Turchin, among others. Taking advantage of rational formality properties of operads that appear in the models yields generalizations of Vassiliev's knot invariants to other spaces of embeddings. New understanding of the role of operads here now suggest some very elegant conjectural explicit deloopings of spaces of high dimensional `long knots'. Another idea has emerged in work of Arone: that one can `play off' Embedding and Orthogonal Calculus to learn more about resolutions of embedding spaces.

At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.

A simple and familiar example of this is the Euler characteristic $e(X)$, where the local-to-global property for good decompositions takes the form $e(U union V) = e(U) + e(V) - e(U intersect V)$. A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer--Vietoris sequence. Finally one can consider the functor $SP^infty: Top --> Top$, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups $pi_*(SP^infty(X)) = H_*(X)$, the Meyer--Vietoris sequence for homology is thus a consequence of applying $pi_*( )$ to the homotopy pullback square.

It was the insight of Tom Goodwillie in the 1980's that such ``linear'' functors $F: Top --> Top$ form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree n takes appropriate sorts of $(n+1)$--dimensional cubical homotopy pushout diagrams to $(n+1)$--dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

In his analysis, an arbitrary homotopy functor $F: Top --> Top$, maps to a ``Taylor tower'' of fibrations

$... ---> P_nF ---> P_1F- --> P_0F$, where $P_nF$ is the best degree n polynomial approximation to F. The layers of the tower are always infinite loopspaces of a special form: for each d there is a spectrum $d_n(F)$ with an action of the nth symmetric group $S_n$, such that the homotopy fiber of $P_dF(X) --> P_{d-1}F(X)$ is naturally equivalent to the zero space of the homotopy orbits of the spectrum with $S_n$ action $(d_n(F)$ smash $X^n)$. In particular, associated to a functor F is a symmetric sequence in spectra ${d_0(F), d_1(F), d_2(F), ...}$.

Two kinds of developments suggest the centrality of Goodwillie's ideas in modern algebraic and geometric topology.

Firstly, functor calculus comes in a remarkable number of varieties: the features just described persist in the study of functors $F: C --> D$ for many interesting pairs of categories C and D. Functor calculus includes Homotopy Calculus in which C and D are Top or Spectra, Orthogonal Calculus in which C is the category of either real or complex inner product spaces, Andre-Quillen Calculus in which C and D are categories of commutative S--algebras or similar, and Embedding Calculus in which C is the category of open subsets of a fixed smooth manifold M.

Secondly, when one looks at explicit examples of basic functors, lovely and unexpected features have emerged: the symmetric sequences are often modeled by important operads, and the Taylor towers often behave well after either rational localization or localization with respect to a periodic homology theory. Thus the methods of functor calculus should inform two of the currently most popular directions in topology: operad theory, particularly as it relates to geometric topology, and chromatic topology, including elliptic cohomology and variants.

II. The Workshop

The most basic purpose of the workshop will be to extend awareness of what the homotopical methods of functor calculus offer to those who study operad theory, and reciprocally, to examine what the methods of operad theory and chromatic localization theory offer to the analysis of Taylor towers.

The workshop will address structural questions:

* How much additional information is necessary to reconstruct a Taylor tower from knowledge of a functor's associated symmetric sequence of derivatives?

* To what extent does this sort of classification simplify after localization?

* How closely is the study of Taylor towers in very general contexts related to the theory of operads and their modules?

* What is the relationship of calculus to Koszul duality for operads?

The workshop will address computational applications:

* The use of Orthogonal Calculus to further our understanding of how unstable homotopy groups of spheres are built out of stable information.

* The use of Homotopy Calculus and Andre-Quillen calculus to extract new information about the homology of iterated loop spaces.

* The use of Embedding Calculus, coupled with the theory of operads, to obtain detailed formulas for homotopy types, homology groups, and homotopy groups of spaces of embeddings.

We expect participants to include the lead researchers working on such problems as well as postdocs and advanced graduate students looking to begin work in these areas. There seems to be enthusiasm for this workshop: we contacted most of the people listed below about their interest in attending, and everyone of this group answered positively within 24 hours.

III. A Survey of Recent Research

G. Arone and M. Mahowald showed the potential of applying Homotopy Calculus to the one of most formidable problems of algebraic topology: the calculation of the homotopy groups of spheres. Much of what is classically known follows from two themes: relate unstable homotopy to stable homotopy, and study localization with respect to periodic homology theories. Historically, the first of these has been the study of Hopf invariants and related spectral sequences, and the second has focused on localization with respect to K--theory and resulting calculations of $``v_1--periodic''$ homotopy. Arone and Mahowald's analysis of the Taylor tower of the identity functor $I: Top --> Top$, evaluated at $S^m$ for m odd, offers insight into both themes that trumped much work of the previous decades in this area. This was the first example illustrating how, serendipitously, calculus seems to fit beautifully with the methods of stable homotopy theory as chromatically stratified by Morava K--theories and associated $v_n--self$ maps of finite complexes.

Another result showing how towers fit with chromatic information is N. Kuhn's theorem that polynomial endofunctors of Spectra split into their homogeneous pieces after localization with respect to any Morava K--theory. Featured here, and in earlier work of R. McCarthy, is the equivariant Tate construction. Kuhn also has detailed results about the Morava K--theory of infinite loop spaces as a consequence of a highly structured special case of this splitting. Here his arguments make use of what might be termed Andre--Quillen Calculus, as it uses the Taylor tower for the identity functor $I: Alg --> Alg$, where Alg is the category of commutative nonunital S--algebras. The linearization of I is a version of Topological Andre--Quillen homology.

M. Weiss's Orthogonal Calculus is an adaptation of the Taylor tower to functors from the domain category of finite-dimensional real or complex inner product spaces. The idea is to reconstruct such a functor from knowledge of its behavior on "large'' vector spaces. For example, in the real case, the functor sending V to BO(V) gives an elegant tower converging to BO(V), with first term BO. Very recent work of Arone, W. Dwyer, and K. Lesh shows that there are strong ties between orthogonal calculus and homotopy calculus, and uses these ties to obtain otherwise inaccessible delooping results for structure maps in the tower studied by Arone and Mahowald. A next generation of results about the homotopy groups of spheres, both classical and periodic, seems on the horizon.

Working within the Homotopy Calculus setting, the recent paper of Arone and M. Ching signals a new level of sophistication with respect to our understanding of how to reconstruct a Tayor tower from the symmetric sequence that determines its layers. Firstly, the symmetric sequence $d_*(I_C)$ of the identity functor on C will be an operad in the category Sp(C) of symmetric C-spectra. For example, when C is Top, one gets a geometric version of the Lie operad. Secondly, given a functor $F: C --> D$, the symmetric sequence $d_*(F)$ will be a right $d_*(I_C)$--module and a left $d_*(I_D)$--module. There is a comparison of the Tayor tower with a tower constructed solely using these operad module structures, with the difference measured by Tate constuctions. Conjecturally this story will generalize to other situations, e.g. that of Andre-Quillen Calculus, where some investigations are ongoing by M. Mandell and M. Basterra. Furthermore, analysis of the suspension functor $C --> Sp(C)$ and its right adjoint seems to suggest beautiful connections with the theory of Koszul duality of operads as developed by B. Fresse and others.

In a more geometric context, let O(M) be the category of open subsets of a smooth manifold M. Given another manifold N, the space of immersions functor, $Imm( ,N): O(M)^{op} --> Top$, sends pushouts to pullbacks. By constrast, the space of embeddings functor, $Emb( ,N)$, has less favorable local-to-global behavior. However, the Embedding Calculus of Goodwillie and Weiss offers a systematic way to pass from immersions to embeddings. There has been interesting work in this area by Arone, V. Lambrechts, I. Volic, D. Sinha, and V. Turchin, among others. Taking advantage of rational formality properties of operads that appear in the models yields generalizations of Vassiliev's knot invariants to other spaces of embeddings. New understanding of the role of operads here now suggest some very elegant conjectural explicit deloopings of spaces of high dimensional `long knots'. Another idea has emerged in work of Arone: that one can `play off' Embedding and Orthogonal Calculus to learn more about resolutions of embedding spaces.