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«Manifolds, K-Theory, and Related Topics Dubrovnik, Croatia, June 23–27, 2014 Titles and abstracts These are the titles and abstracts for all the ...»

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Manifolds, K-Theory, and Related Topics

Dubrovnik, Croatia, June 23–27, 2014

Titles and abstracts

These are the titles and abstracts for all the conference talks, plenary and contributed. The speakers are listed alphabetically.

David Ayala (Montana State)

Title: On the Taylor tower of factorization homology

Abstract: Factorization homology is a ‘coordinate free’ pairing of an n-manifold and an n-Disk algebra – it specializes to topological

Hochschild homology for the case of a circle. I will explicate the Goodwillie cofiltration of factorization homology in terms of configuration spaces, and argue that it converges on an understood class of algebras. From this we can deduce a general formulation of Poincar´ duality as it intertwines with Koszul duality. This duality accommodates some easy e calculations, after which we can make some open observations about a duality among TQFT’s that exchanges perturbative and non-perturbative. This is joint work with John Francis.

David Barnes (Queen’s University) Title: Rationalising the orthogonal calculus Abstract: Orthogonal calculus is a calculus of functors, inspired by Goodwillie calculus. It takes as input a functor from finite dimensional inner product spaces to topological spaces and as output gives a tower of approximations by well-behaved functors. The output captures a lot of important homotopical information and is an important tool for calculations.

This talk will describe current work on setting up model structures for a rational version of Weiss’ orthogonal calculus.

Irina Bobkova (Northwestern) Title: Towards a resolution of the K(2)-local sphere at the prime 2 Abstract: The homotopy of the p-local sphere spectrum S is determined by a family of localizations LK(n) S with respect to Morava K-theories K(n). We will discuss some computations when p, n = 2. Considerable information here can be derived from the action of the Morava stabilizer group on the Lubin-Tate theory. Goerss, Henn, Mahowald and Rezk have constructed a resolution of the K(2)-local sphere at the prime 3 which allows to simplify computations of π∗ LK(2) S. We will discuss a generalization of their work to the prime 2 and construct a resolution of the spectrum E hS2, which is closely related to the K(2)-local sphere at the prime 2.

Gunnar Carlsson (Stanford) Title: On the topology of finite metric spaces Abstract: Finite metric spaces are an excellent way to model problems coming from the analysis of large and complex data sets.

Over thelast 15 years, there has been a development of cohomological methods for dealing with them. I will survey the results and future directions for this work.

Wojciech Chacholski (KTH) Title: Idempotent symmetries of spaces Abstract: This is a joint work with E. Dror Farjoun, R. Flores, and J. Scherer.

Let G be a group. Define BG to be the collection of these connected spaces X for which π1 X = G and the map π1 : Map∗ (X, X) → Hom(G, G) is a weak equivalence. The Eilenberg-Mac Lane space K(G, 1) clearly belongs to BG.

The question is if there are other spaces in BG. Flores and Scherer showed that BΣ3 contains a space whose universal cover is the homotopy fiber of the degree 3 map on the sphere S 3. During the talk I will present evidence for the following

conjecture:

If G is a finite group, then BG contains only finitely many different homotopy types.

This conjecture is a reflection of certain symmetries of the category of pointed spaces. An augmented functor of spaces cX : φ(X) → X is called idempotent if Map∗ (φ(X), cX ) : Map∗ (φ(X), φ(X)) → Map∗ (φ(X), X) is a weak equivalence for any X. The A–cellularization cA,X : CWA X → X is a typical example. The conjecture is equivalent to the orbit under the idempotent functors of the Eilenberg-Mac Lane space K(G, 1) being finite for a finite G. For evidence for the conjecture one mighty look at orbits of idempotent deformations in other categories. For example it is a theorem that in the category of groups, a finite group has only finitely many idempotent deformations. Furthermore the orbit of a finite simple group can have at most 7 elements. In the talk I will explain why the conjecture is true if G is nilpotent and present some of its consequences. One of the steps to prove that case is a generalization of the Blakers–Massey theorem.

Stanley Chang (Wellesley) Title: The virtual structure set Abstract: This talk will introduce the notion of a structure set in the context of the surgery exact sequence. We will show that the Borel conjecture for compact manifolds cannot be extended into the proper noncompact setting by exhibiting examples of arithmetic manifolds whose proper structure set is nontrivial. At the end we will also introduce the notion of a virtual structure set defined on an infinite sequence of covers, and explain some interesting calculations regarding these objects.

Michael Ching (Amherst) Title: Manifolds, K-theory and the calculus of functors Abstract: I will explain joint work with Greg Arone that describes the structure on the derivatives needed to reconstruct the

Goodwillie tower of a homotopy functor from based spaces to spectra. For some functors, this structure has a nice form:

the derivatives form a module over the Koszul dual of the stable little n-discs operad. We can characterize the functors for which this is the case as those that are left Kan extensions from a certain category of ”pointed framed n-dimensional manifolds” and ”pointed framed embeddings”. I will also try to describe joint work with Greg Arone and Andrew Blumberg to figure out where Waldhausen’s algebraic K-theory of spaces fits into this picture.





Boris Chorny (Haifa) Title: A classification of small linear functors Abstract: We extend Goodwillie’s classification of finitary linear functors to arbitrary small functors. (A functor is small if it commutes with λ-filtered colimits for some cardinal λ.) Namely we show that every small linear simplicial functor from spectra to simlpicial sets is weakly equivalent to a filtered colimit of representable functors represented in cofibrant spectra. Moreover, we present this classification as a Quillen equivalence of the category of small functors from spectra to simplicial sets equipped with the linear model structure and the opposite of the pro-category of spectra with the strict model structure.

Ralph Cohen (Stanford) Title: Comparing Topological Field Theories: the string topology of a manifold and the symplectic cohomology of its cotangent bundle Abstract: I will describe joint work with Sheel Ganatra, in which we prove an equivalence between two chain complex valued topological field theories: the String Topology of a manifold M, and the Symplectic Cohomology of its cotangent bundle, T ∗ M. I will also discuss how the notion of Koszul duality appears in the study of TFT’s.

Emanuele Dotto (MIT) Title: Homotopy theory of equivariant diagrams and equivariant calculus of functors Abstract: In recent joint work with Kristian Moi we develop a homotopy theory of equivariant diagrams. Given a finite group G acting on a category I, we consider I-shaped diagrams in a model category C (typically spaces or spectra) that are equipped with a “G-action that permutes the verticies of the diagram”. When I = P (J) is the poset of subsets of a finite G-set J, we call these diagrams equivariant cubes. The Bousfield-Kan formula provides homotopy (co)limit functors form the category of equivariant diagrams in C to the category of G-objects in C. For equivariant cubes this leads to a notion of homotopy (co)cartesian G-cubes. A functor is G-excisive if it sends cocartesian G-cubes to cartesian G-cubes.

I will relate this notion of equivariant excision to equivariant cohomology theories and to previous work on G-linearity of Blumberg. If time allows, I will talk about higher G-excision and how to set up an equivariant Goodwillie Taylor tower, and discuss convergence for functors that satisfy an equivariant Blakers-Massey theorem.

Bjørn Dundas (Bergen) Title: Higher topological Hochschild homology Abstract: In the 1980’s Goodwillie conjectured the existence of a Hochschild-style theory agreeing with stable K-theory. This proved to be a very fruitful point of view; refinements and generalizations have both provided calculations of important invariants and access to new ones. Most importantly, it has given us an inroad to questions about algebraic K-theory through stable equivariant homotopy theory. In this talk I will discuss some very recent calculations j/w Lindenstrauss and Richter of higher topological Hochschild homology.

Rosona Eldred (M¨nster) u Title: Goodwillie calculus and nilpotence Abstract: Viewing spectra as the homotopy-abelianization of spaces, the Goodwillie taylor tower of a homotopy functor is a sort of homotopy-nilpotent tower. Work of Biedermann-Dwyer shows that (loops on) n-excisive functors land in spaces which are homotopy-nilpotent in a related sense. I will discuss a strengthening of this result by establishing a stronger property on the finite limits used to construct the polynomial approximations. Time permitting, I will also describe the dual setting, which involves spaces with bounded cup product length, and related conjectures of further structure.

Tom Goodwillie (Brown) Title: Scissors congruence in mixed dimensions Abstract: We define a graded ring E in which En is a Grothendieck group for compact polytopes of dimension at most n in Euclidean spaces. (The cokernel of the evident map En−1 → En is the nth Euclidean scissors congruence group.) Ring maps E → R are called multiplicative invariants with values in the ring R; the first examples are the volume E → R and the Euler characteristic χ : E → Z. We consider also an analogous object L, which is based not on polytopes but on germs of polytopes at a point. The ring L has more structure than E; in particular it is a kind of Hopf algebroid. This means that there is a partial multiplication of local multiplicative invariants L → R. (Its definition generalizes that of the Dehn invariant.) In fact, Spec L can be viewed as the morphism space of a groupoid scheme over Z whose object space is the affine line. The local invariants act on the global invariants. Starting with obvious invariants related to volume and Euler characteristic, one can then create a family of invariants with values in polynomial rings over tensor powers of the real numbers.

Jesper Grodal (Copenhagen) Title: The gluing conjecture Abstract: I will discuss progress on the glueing conjecture in block theory, including a verification for finite groups of Lie type. This involves looking at finite groups of Lie type through the eyes of homotopy theory.

Ian Hambleton (McMaster) Title: Finite group actions and chain complexes over the orbit category Abstract: The unit spheres in orthogonal representations of finite groups give examples of group actions on spheres. We investigate non-linear actions by studying chain complexes over the orbit category, and constructing finite G-CW complexes. This leads to new examples of homotopy representations with isotropy of rank one. This project is joint with Ergun Yalcin (Bilkent University, Ankara).

Kathryn Hess (Lausanne) Title: Waldhausen K-theory via comodules Abstract: I will present joint work with Brooke Shipley, in which we have defined a model category structure on the category of Σ∞ X+ -comodule spectra such that the K-theory of the associated Waldhausen category of homotopically finite objects is naturally weakly equivalent to the usual Waldhausen K-theory of X, A(X), when X is simply connected. I will sketch a number of application of this new approach to the K-theory of spaces and describe its relation to the more familiar description in terms of Σ∞ ΩX+ -module spectra.

Sadok Kallel (American University Sharjah) Title: On the topology of diagonal arrangements and their complements Given a finite simplicial complex X, a diagonal arrangement in X n is the union of various ”diagonal subspaces” in this

Abstract:

product. The complement of these arrangements give a generalization of the standard configuration space construction.

In this talk we discuss the homology and homotopy groups through a range of these diagonal arrangements, their stable splittings and Euler characteristics. The answers are given in terms of invariants of X.

John Klein (Wayne State) Title: Topological Stochastics Abstract: This talk is about statistical field theories arising from observables which are homological in nature. Starting with a finite CW complex of finite dimension d, I will show how to construct a stochastic process in which the state space is given by integer-valued cellular (d-1)-cycles. A trajectory is given by a sequence of states equipped with waiting times, in which successive states are by joined by instantaneous jumps over d-cells.

I will then construct a family of current observables and show how it gives rise to a real homology class in degree d called, ”average current.” Lastly, I’ll discuss a fractional quantization result, which roughly says that if certain parameters driving the system (driving time, inverse temperature) tend to infinity, then the average current converges to a rational homology class.

Ben Knudsen (Northwestern) Title: Configuration spaces and factorization homology Abstract: I will describe recent work using factorization homology to compute the Betti numbers of the configuration spaces of k unordered points in an arbitrary manifold, possibly with boundary, generalizing theorems of Bodigheimer-Cohen-Taylor and Felix-Thomas.

Robin Koytcheff (Victoria) Title: Homotopy BottTaubes integrals and the Milnor triple linking number Abstract: Bott and Taubes produced knot invariants by considering a bundle over the space of knots and integrating differential forms along its fiber. Their methods were used to construct all Vassiliev (or finite-type) knot invariants, as well as Vassiliev-type classes, which are real cohomology classes in spaces of knots in Euclidean space of dimension at least 4.



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