Topology Ecuador 2017

Based on joint work with Paolo Antonini and Georges Skandalis.

---

Lauren Bandklayder, Northwestern University

A proof of the Dold-Thom theorem using factorization homology

The Dold-Thom theorem is a classical result in algebraic topology relating homotopy and homology. It states that for a nice, based topological space,M, there is an isomorphism between the homotopy groups of the symmetric product of M and the homology groups of M itself: π∗(Sym(M;A))≅H∗(M;A). The crux of most known proofs of this is to check a certain map is a quasi-fibration. This is a bit of a technical digression, and it my goal present a more direct proof which does not require any such fact.

---

Jacek Brodzki, University of Southampton

A differential complex for CAT(0) cubical spaces

In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. These constructions and ideas have foundational applications to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p adic groups.
This talk will present an extension of the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. The construction is quite intricate and relies in a crucial way on the rich and beautiful geometry of these spaces. To illustrate the utility of the extended construction we will sketch an application of these ideas to operator K-theory where we give a new proof of K-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces. This is joint work with Nigel Higson and Erik Guentner.

---

Bob Bruner, Wayne State University and University of Oslo

A(2)-Modules and their Cohomology

Modules over subalgebras of the Steenrod algebra (such as A(2) exhibit periodicities and localizations which illuminate their module structure and simplify the calculation of their cohomology.

This has topological correlates. For example, just as

• A(2) modules are related to topological modular forms,
• A(1) modules are related to real K-theory, and
• A(0) modules to integral cohomology.

We will discuss these and related theories and the manner in which these periodicities appear as the complexity of the subalgebra increases.

---

Moira Chas, Stony Brook University

Computer driven questions, pre-theorems and theorems in geometry

Several numbers can be associated to free homotopy class X of closed curves on a surface S with boundary and negative Euler characteristic. Among these,

• the self-intersection number of X
• the word length of X
• the length of the geodesic corresponding to X
• the number of free homotopy classes of a given word length the mapping class group orbit of X

The interrelations of these numbers exhibit many patterns when explicitly determined or approximated by running a variety of algorithms in a computer.
We will discuss how these computations lead to counterexamples to existing conjectures and to the discovery of new patterns . Some of these new patterns, so intricate and unlikely that they are certainly true (even if not proven yet), are "pre-theorems". Many of these pre-theorems later became theorems. An example of such a theorem states that the distribution of the self-intersection of free homotopy classes of closed curves on a surface, appropriately normalized, sampling among given word length, approaches a Gaussian when the word length goes to infinity. An example of a counterexample (no pun untended!) is that there exists pairs of length equivalent free homotopy classes of curves on a surface S that have different self-intersection number. (Two free homotopy classes X and Y are length equivalent if for every hyperbolic metric M on S, M(X) = M(Y)).

This talk will be accessible to grad and advanced undergrad students.

---

Cesar del Corral, Universidad de los Andes

Noncommutative Residue and Canonical Trace on a Manifold with Boundary

This talk addresses the existence (and uniqueness) of the noncommutative residue and canonical trace on pseudo-differential operators on manifolds with boundary.

On the one hand, the noncommutative residue was introduced in 1984 by Wodzicki M. and independently by Guillemin V., the authors shown that the noncommutative residue is the unique trace on the class of classical pseudo-differential operators on a closed manifold, i.e. the unique linear functional which vanishes on commutators of operators. This functional was popularized by A. Connes in the context of the noncommutative geometry because of its many application in such field, as in many others. On the other hand, the canonical trace on closed manifolds was studied and popularized by Kontsevich, M. and Vishik, S. 1995, and shown to be the unique trace on appropriate classes of classical pseudo-differential operators which extends the usual trace for trace class operators.

Fedosov, B.V., Golse, F., Leichtnam, E. and Schrohe, E. provided an extension of the noncommutative residue on an algebra of boundary value operators for a manifold with boundary, called the Boutet de Monvel algebra. Such algebra exclude the class of operators where the canonical trace is defined, namely, operators of non-integer order. Drawn by the necessity to circumvent the transmission property we provide a class classical pseudo-differential operator on a manifold with boundary (which contains the Boutet de Monvel algebra) and extends the canonical trace to this class. Finally, we discuss as the traciality property as the uniqueness of the noncommutative residue and the canonical

trace in the case of a manifold with boundary, and we give some possible new applications to noncommutative geometry and C*-algebras following the approaches from Connes A., Scott S., Schrohe, E. and many others authors.

---

Soren Galatius, Stanford University and University of Copenhagen

Homological stability and meta-stability for mapping class groups of surfaces

Let Γg,1 denote the mapping class group of a genus g surface with one parametrized boundary component. The group homology Hi(Γg,1) is independent of g, as long as g is large compared to i, by a famous theorem of Harer known as homological stability, now known to hold when 2g>3i. Outside that range, the relative homology groups Hi(Γg,1,Γg−1,1) contain interesting information about the failure of homological stability. In this talk, I will discuss a "metastability" result, obtained in joint work with Oscar Randal-Williams and Alexander Kupers: these relative homology groups depend only on the number k=2g−3i, as long as g is large compared to k.

---

José Manuel Gómez, Universidad Nacional de Colombia, sede Medellín

Commutative K-theory

In this talk I will introduce a new generalized cohomology theory called Commutative K-theory.

This theory is defined transitionally commutative vector bundles.

I will start the talk by introducing the space of commuting elements in a Lie group
G and discuss a few simple examples. Then I will show how those spaces can be assembled together obtain a space that is called the classifying space for commutativity in G and how these spaces classify precisely transitionally commutative vector bundles.

After that I will introduce commutative K-theory and discuss some general properties.

I will also discuss the recent computation of the homotopy groups of this theory made by Simon Gritschacher.

This is a joint work with Alejandro Adem, John Lind and Ulrike Tillmann.

---

John Greenlees, University of Sheffield

The structure of rational equivariant cohomology theories

If one wants to understand modules over a field one can classify them, but in general the way to understand categories of modules is to think of them as sheaves over the prime spectrum. The same thing applies to rational equivariant cohomology theories for a compact Lie groupG. For finite groups one may classify them, but in general one should think of them as sheaves over the Balmer spectrum. The talk will describe the Balmer spectrum (its points correspond to closed subgroups of G) and the Zariski topology on it.

In principle the talk could continue to describe the nature of the algebraic model this engenders, but in practice there is unlikely to be time for this.

The key ingredients are the Borel-Hsiang-Quillen localization theorem (identifying the rational cohomology of fixed points under a torus for a finite G-space), tom Dieck's identification of the Burnside ring and the algebraic model of free cohomology theories due to the speaker and Shipley.

---

Fabian Hebestreit, Universität Bonn

A vanishing theorem for tautological classes of aspherical manifolds
Joint with M. Land, W. Lück and O. Randal-Williams

Tautological or generalised Morita-Miller-Mumford classes have recently been of great interest due to the work of Galatius and Randal-Williams who used them to describe the cohomology of classifying spaces of diffeomorphism groups of many high dimensional, high-genus manifolds in a range. In contrast, the diffeomorphism groups of high dimensional aspherical manifolds have long been studied successfully using surgery and Waldhausen's A-theory. However, since the genus of an aspherical manifold vanishes previous work reveals little about their tautological classes.

In the talk I will explain a recent result of ours which shows that on almost all aspherical manifolds almost all tautological classes almost vanish. Two of these almosts are enforced by hyperbolisation and the Madsen-Weiss theorem, respectively, whereas the third is an artefact of our methods, which rely on both the Farrell-Jones conjecture and a less known conjecture of Burghelea in group cohomology.

---

Kathryn Hess, Ecole polytechnique fédérale de Lausanne

Topology meets neuroscience

I will present an overview of applications of topology to neuroscience on a wide range of scales, from the level of neurons to the level of brain regions. In particular I will describe collaborations in progress with the Blue Brain Project on topological analysis of the structure and function of digitally reconstructed microconnectomes and on topological classification of neuron morphological types. I will then briefly sketch applications of topology to the analysis of brain imaging data.

---

Ralph Kaufmann, Purdue University

Moduli Spaces and Feynman categories

Feynman categories provide a unifying categorical framework for several aspects appearing in algebraic topology, non-commutative geometry and moduli spaces.

There is combinatorial level which describes such things as operads and modular operads as functors from a category whose morphisms have a graph theoretical presentation. Through resolutions and a W-constructions, this combinatorial level is elevated to a dg-setting or to a topological setting. Finally, non-(co)commutative bi and Hopf algebras naturally appear in this context as well and provide a link to motivic constructions.

We will shortly give the axiomatic setup and then focus on the construction of moduli spaces using the W-construction and decorations.

If time permits, we will also briefly discuss the Hopf algebras and possible implications including multiple zeta values and Grothendieck-Teichmueller theory.

---

Ben Knudsen, Harvard University

Subdivisional spaces and graph braid groups

We develop an approach to the study of the configuration spaces of a cell complex X that is both flexible and suitable for computation. We proceed by viewing X, together with its subdivisions, as a "subdivisional space," a kind of diagram object, which has associated to it certain diagrammatic versions of configuration spaces. These objects, which model the correct homotopy types, mix the discrete and the continuous, and they may be attacked by combining techniques drawn from discrete Morse theory and factorization homology. We apply our theory in the 1-dimensional example of a graph, obtaining a new family of chain models for the homology of graph braid groups depending only on intrinsic graph theoretic data. These "intrinsic complexes" come equipped with a robust computational toolkit, which we exploit in numerous calculations, old and new. This is joint work with Byung Hee An and Gabriel Drummond-Cole.

---

Kathryn Lesh, Union College

A classification of p-toral subgroups with noncontractible fixed point sets on decomposition space

Recent joint work with G. Arone combines with joint work with J. Bergner, R. Joachimi, V. Stojanoska, and K. Wickelgren to give a complete classification of p-toral subgroups of U(n) that have noncontractible fixed points on the the poset of decompositions of complex n space into proper orthogonal subspaces.

---

Ernesto Lupercio, Cinvestav IPN Mexico

Quantum Toric Varieties

In this talk I will explain our generalization of toric varieties to the non-commutative geometric realm. The resulting quantum toric varieties are built up out of quantum tori and classical toric varieities are an example. Unlike the classical counterpart, quantum toric varieties have moduli.

---

Peter May, University of Chicago

Glimpses of equivariant algebraic topology

From P.A. Smith to the present. Around 80 years ago, Smith proved the remarkable result that if a finite p group G acts on a compact space X having the mod p homology of a sphere, then the fixed point space XG also has the mod p homology of a sphere. Equivariant algebraic topology has developed in fits and starts ever since. I'll give some glimpses of current directions and questions.

---

Jovana Obradovic, Université Paris Diderot and Inria, France

Categorified cyclic operads

The goal of this work is to establish the notion of categorified cyclic operad. This is done by weakening the axioms of cyclic operads, from equalities to natural isomorphisms, and by formulating conditions concerning these isomorphism which ensure coherence. For entries-only cyclic operads, this coherence is of the same kind as the coherence of symmetric monoidal categories: all diagrams made of associator and commutator isomorphisms are required to commute. However, in the setting of cyclic operads, where the existance of objects and morphisms depends on the shape of a fixed unrooted tree, these arrows do not always exist. In other words, the coherences that Mac Lane established for symmetric monoidal categories do not solve the coherence problem of categorified cyclic operads. We exhibit the appropriate conditions of this setting and prove the coherence theorem, relying on a result of Dosen and Petric coming from the coherence of categorified operads. Additionally, by the equivalence between the two possible characterisations of cyclic operads, for cyclic operads introduced as operads with extra structure (that exchanges the output of an operation with one of its inputs), i.e. for exchangeable-output cyclic operads, we examine which of the axioms of the extra structure needs to be weakened (in order to lift that equivalence to weakened structures), and we exhibit the appropriate coherence conditions.

---

Kevin Piterman, Universidad de Buenos Aires

The homotopy types of the posets of p -subgroups

The posets Sp(G) and Ap(G) of non-trivial p-subgroups and elementary abelian p-subgroups of a finite group G were introduced by K. Brown and D. Quillen in the 70's to study the connections between their topological properties and the algebraic properties of the group. A very well-known conjecture of Quillen states that the simplicial complex associated to Sp(G) is contractible if and only if G has a non-trivial normal p-subgroup.

Quillen's conjecture is still open in general, but there have been considerable advances in this direction in the last four decades, especially by the work of Aschbacher and Smith. In the 80's R.E. Stong investigated the posets Sp(G) and Ap(G) from the point of view of the theory of finite topological spaces, and a few years ago J. Barmak reformulated Quillen's conjecture using simple homotopy theory of finite spaces. In this talk I will recall some of the most relevant results on these posets and exhibit new results that we obtained recently in collaboration with Gabriel Minian. In particular I will describe the contractibility of the finite space Ap(G) in terms of the algebraic properties of G

---

Kate Poirier, City University of New York - New York City College of Technology

Moduli spaces of graphs for string topology

String topology studies natural operations on the loop space of a manifold arising from intersections of loops. One wishes to have an understanding of a universal space parametrizing all such operations. Moduli spaces of Riemann surfaces--and their compactifications--are known to be helpful in this parameterization, usually via combinatorial models using graphs which capture intersection information. Different treatments of string topology offer different spaces of graphs which have different relationships with moduli space and with one another. Egas and Kupers provide one attempt at synthesizing the relationship among these spaces of graphs by comparing certain combinatorial models of moduli space that have been used for string topology. In this talk, we provide another chapter of this synthesis by describing the relationship between the spaces of graphs parametrizing the chain-level string topology operations of Drummond-Cole, Poirier, and Rounds and the spaces of graphs parameterizing the algebraic string operations of Tradler and Zeinalian. We also describe how they fit into the Egas-Kupers and moduli space pictures.

---

Manuel Rivera, University of Miami - CINVESTAV

Cubical rigidification, cobar construction, based loop space

I will explain how to generalize Adams' cobar construction to possibly non-simply connected spaces. The methods used, which are of independent interest, consist on a cubical intepretation of the Lurie rigidification functor originally definied to compare two models of infinity-categories.

---

Claudia Scheimbauer, Oxford University

Fully extended functorial field theories and dualizability in the higher Morita category

Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the topic.

Natural targets for extended topological field theories are higher Morita categories: generalizations of the bicategory of algebras, bimodules, and homomorphisms.

After giving an introduction to topological field theories, I will explain how one can use geometric arguments to obtain results on dualizablity in a "factorization version" of the Morita category and using this, examples of low-dimensional field theories "relative" to their observables. An example will be given by polynomial differential operators, i.e. the Weyl algebra, in positive characteristic and its center. This is joint work with Owen Gwilliam.

---

Jérôme Scherer, Ecole polytechnique fédérale de Lausanne

Relative resolutions for unbounded chain complexes

This is joint work with W. Chacholski, A. Neeman, and W. Pitsch. Sometimes it is difficult (or maybe even unknown) to construct a model structure in a specific homotopical situation. This is so for the category of unbounded chain complexes over a commutative ring where the weak equivalences should reflect some relative properties (by changing the class of injective modules say). This is when a model approximation comes in handy because it provides more flexibility than a model structure. I will explain what a model approximation is and show that Spaltenstein's idea of truncating a complex, resolving the pieces, and taking the limit, provides such a model approximation, at least when the category of chain complexes satisfies a relative version of Roos' axiom AB4*-n. This axiom is satisfied in many interesting examples, but fails sometimes, as should be expected ...

---

Tomer Schlank, Hebrew University In Jerusalem

Ultra-Products and Chromatic Homotopy Theory

Let Cp,n be the E(n)-local category at height n and prime p. These categories are of great interest to the stable homotopy theorist since they serve as a the "bounded pieces" of the chromatic filtration on the category of spectra. It is a well known observation that for a given height n certain "special" phenomena happen only for small enough primes. Further, in some sense, the categories Cp,n become more regular and algebraic as p goes to infinity for a fixed n. The goal of this talk is to make this intuition precise.

In a 96' unpublished paper Jens Franke defined for every height n and a prime p the category Frn,p of Quasi-periodic complexes on the stack MFG(p)≤n as an algebraic analog of Cp,n. Now given an infinite sequence of mathematical structures, logicians have a method to construct a limiting one by using ultra-products. We shall define a notion of ultra-product of categories and then for a fixed height n we prove:

                                      Ultra                           Ultra
                                     ∏          Cn,p≅            ∏    Frn,p.
                                     p                                 p

This is a joint project with N. Stapleton and T. Barthel.

---

Paul Arnaud Songhafouo Tsopméné, University of Regina, Canada

Generalization of Manifold Calculus

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors

F:O(M)→Top from O(M) to the category of spaces (of which the embedding functor Emb(-,W) for a fixed manifold W is a prime example). Weiss showed that polynomial functors of degree ≤k are determined by their values on Ok(M), where Ok(M) is the full subposet of O(M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace Ok(M) by more general subposets and still recover the same notion of polynomial functor. In this talk, we will first explain the philosophy of Manifold Calculus. Next we will give some examples and applications. Lastly, we will show that the aforementioned results can be generalized to functors from

---

Dennis Sullivan, Stony Brook University - CUNY Graduate Center

Combinatorial Hydrodynamics

The motion of real fluids can only approximate the motion described by the PDEs that model fluid mechanics. For example the beautiful Euler PDE for incompressible zero viscosity [no friction] fluid evolution says the twisting or turning in 3D of the motion [the vorticity] is continually transported forward by the motion. In topology one approximates the spatial continuum by a cell complex. Using the ideas that are needed to construct the induced transformation in combinatorial topology we construct a cellular or combinatorial model of " the vorticity is transported by the motion". We will report on some studies of this model and how they relate to physical incompressible fluid motion.

---

Christian Wimmer, University of Bonn

Rational equivariant commutative ring spectra

Rational global homotopy theory
Stable homotopy theory simplifies drastically if one consider spectra up to rational equivalence.

It is well-known that taking homotopy groups induces an equivalence

                                     G-SHC  ≃Qgr.         ∏            Q[WH]-mod
                                                                 (H≤G)

between the genuine G-equivariant stable homotopy category (G finite) and the category of graded modules over the Weyl groups WH indexed by the conjugacy classes of subgroups of G.

However, this approach is too primitive to be useful for the comparison of highly structured ring spectra in this setting.

I will explain how geometric fixed points equipped with additional norm maps related to the Hill-Hopkins-Ravenel norms can be used to give an algebraic model.

They induce an equivalence

                              Com(G-Spectra)Q≃OrbG-CDGAQ

between the homotopy theory of rational commutative G-ring spectra and OrbG-diagrams in rational commutative differential graded algebras, where OrbG is the orbit category of the group G. I will also try to indicate the analogous global equivariant statements.

Global homotopy theory has recently been extensively developed by Schwede. It is a framework that makes sense of homotopy types encoding compatible actions by all (say finite) groups. As in more classical situations, rationally the global homotopy category simplifies. However in contrast to genuine G-spectra it does not split. I will discuss a rather explicit equivalence to an easily defined derived category via a geometric fixed points construction.